KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT BURGERLIJKE BOUWKUNDEKasteelpark Arenberg 40, B-3001 Leuven

THE IMPACT OF UNCERTAIN

DYNAMIC SOIL CHARACTERISTICS

ON THE PREDICTION

OF GROUND VIBRATIONS

Promotoren: Proefschrift voorgedragen totProf. dr. ir. G. Degrande het behalen van het doctoraatProf. dr. ir. G. De Roeck in de ingenieurswetenschappen

door

Mattias Schevenels

Mei 2007

KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT BURGERLIJKE BOUWKUNDEKasteelpark Arenberg 40, B-3001 Leuven

THE IMPACT OF UNCERTAIN

DYNAMIC SOIL CHARACTERISTICS

ON THE PREDICTION

OF GROUND VIBRATIONS

Jury: Proefschrift voorgedragen totH. Van Brussel, voorzitter het behalen van het doctoraatG. Degrande, promotor in de ingenieurswetenschappenG. De Roeck, promotorJ.-P. Coyette (Universite Catholique de Louvain) doorG. LombaertD. Vandepitte Mattias Schevenels

T. Camelbeeck (Universite Libre de Bruxelles)D. Clouteau (Ecole Centrale Paris)A. Metrikine (Technische Universiteit Delft)

U.D.C. 519.21:624.131.38:624.131.55

Mei 2007

© Katholieke Universiteit Leuven – Faculteit IngenieurswetenschappenArenbergkasteel, B-3001 Leuven (Belgium)

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All rights reserved. No part of the publication may be reproduced in any form byprint, photoprint, microfilm, or any other means without written permission fromthe publisher.

D/2007/7515/46ISBN 978-90-5682-812-7

Voorwoord

Bij het realiseren van deze doctoraatsthesis heb ik kunnen rekenen op de hulp vanheel wat mensen. Ik wil hen daarvoor graag bedanken.

Mijn dank gaat in de eerste plaats uit naar Geert Degrande, mijn promotor. Hijheeft mijn interesse in dit onderzoeksdomein opgewekt en het onderzoek in goedebanen geleid. Hij heeft me tegelijk voldoende vrijheid gegeven en bijgestuurdwaar nodig. Verder heb ik hem mogen vergezellen op een aantal internationalecongressen, in Parijs, Lissabon, en Los Angeles. Het waren niet alleen zinvollemaar ook aangename ervaringen, zowel binnen als buiten de werkuren.

Ook Guido De Roeck wil ik graag bedanken. Hoewel hij minder nauw betrokkenwas bij het onderzoek heeft hij toch steeds een grote interesse getoond. Netals Geert Degrande heeft hij mij de kans en het vertrouwen gegeven een aantaloefenzittingen te verzorgen, wat ik met heel veel plezier gedaan heb.

Geert Lombaert wil ik in het bijzonder bedanken voor de hulp die ik van hemheb gekregen, de talloze tips die hij me heeft gegeven, en de tijd die hij voor mijheeft vrijgemaakt. Ik heb onze samenwerking altijd heel aangenaam gevonden, ophet departement, tijdens cursussen en congressen in Parijs, Rome, Albuquerque,en Los Angeles, en op talloze meetcampagnes. Minstens evenveel heb ik genotenvan de uitstapjes naar exotische locaties zoals de Jemez Mountains, Tivoli, hetpaleis van Nero, en la Butte aux Cailles.

Naast Geert Degrande, Guido De Roeck, en Geert Lombaert dank ik DirkVandepitte en Jean-Pierre Coyette voor de begeleiding van dit doctoraatsproject,de interesse in mijn werk, en de constructieve opmerkingen bij de tekst. Verderwil ik ook de andere leden van de jury bedanken, met name Thierry Camelbeeck,Didier Clouteau, en Andrei Metrikine. Hendrik Van Brussel dank ik voor hetvervullen van de voorzitterstaak.

Ongeveer vier jaar heb ik een bureau gedeeld met David Dooms. Zowelop het bureau als daarbuiten kon ik met hem altijd praten over de meestuiteenlopende onderwerpen, al dan niet werkgerelateerd. Daarnaast heeft hijsteeds een grote interesse getoond in mijn onderzoek, ondanks het feit dat onzeonderzoeksdomeinen relatief ver van elkaar liggen. Dat heeft me veel deugd gedaanen daarom wil ik hem graag bedanken.

Stijn Francois dank ik voor het fijne gezelschap tijdens de congressen endaaropvolgende reisjes in Lissabon en Californie, de leerzame discussies over

iii

iv Voorwoord

golfvoortplanting en grond-structuurinteractie (alsook minder serieuze onderwer-pen), en de nuttige maar vooral aangename samenwerking bij het ontwikkelen vaneen aantal MATLAB-toolboxen die aan de basis liggen van dit doctoraatsproject.

Verder bedank ik de overige leden van de afdeling Bouwmechanica, met nameDanielle, Daan, Hamid, Edwin, Shashank, Jaime, Ozer, Ali, Kai, Eliz-Mari, enBram voor de leuke sfeer op het werk en tijdens de activiteiten buitenshuis.

Ik dank het Fonds voor Wetenschappelijk Onderzoek Vlaanderen (FWO-Vlaanderen) dat me gedurende vier jaar als aspirant heeft tewerkgesteld enheeft bijgedragen aan de financiering van mijn deelname aan enkele buitenlandsecongressen.

Tenslotte wil ik graag mijn familie en vrienden bedanken voor de interesse dieze steeds getoond hebben. Mijn ouders wil ik in het bijzonder bedanken voor deonvoorwaardelijke steun die ik altijd heb mogen ervaren.

Mattias SchevenelsLeuven, 8 mei 2007

De invloed van onzekere

dynamische

grondkarakteristieken op de

voorspelling van trillingen

Situering van het onderwerp

Trillingen in de bebouwde omgeving zijn een vorm van milieuhinder die steeds meeraandacht krijgt. Mogelijke bronnen van trillingshinder zijn weg- en treinverkeer,bouwwerkzaamheden zoals het inheien van palen, en industriele machines zoalsweefgetouwen en drukpersen. Deze trillingen kunnen aanleiding geven tot hindervoor personen [27, 50, 105, 207], storing van gevoelige apparatuur [81, 208], enschade aan gebouwen [51, 206].

Trillingen in gebouwen ontstaan ten gevolge van dynamische krachten dieaangrijpen op structuren (de bron) zoals wegen, sporen, tunnels, palen,...Deze krachten worden doorgegeven aan de grond waar ze elastodynamischegolven veroorzaken. De golven in de grond interageren met de funderingenvan nabijgelegen gebouwen (de ontvanger) waar ze structurele trillingen enherafgestraald geluid genereren.

De voorspelling van trillingen in gebouwen is een dynamisch grond-structuurinteractieprobleem dat bestaat uit drie subproblemen: de karakteriseringvan de bron, de transmissie van golven door de grond, en de interactie van deontvanger en het invallend golfveld. Dit probleem wordt meestal in twee stappenopgelost. Eerst wordt een bronmodel gebruikt voor de voorspelling van detrillingen in het vrije veld. Daarna wordt een ontvangermodel gebruikt waarindeze trillingen opgelegd worden als een invallend golfveld.

Recent zijn verschillende bron- en ontvangermodellen ontwikkeld voor devoorspelling van trillingen in de bebouwde omgeving [39, 132, 136, 143, 165].Deze modellen kunnen gebruikt worden om het effect van trillingsreducerendemaatregelen na te gaan, zowel in nieuwe als in bestaande situaties. De

v

vi Samenvatting

modellen zijn gebaseerd op een subdomeinformulering voor dynamische grond-structuurinteractie [16, 36]. De bron (weg, spoor, tunnel, paal,...) en de ontvanger(gebouw) worden meestal gemodelleerd met de eindige-elementenmethode. Degrond wordt gemodelleerd met de randelementenmethode [25, 54]. De randele-mentenmethode is gebaseerd op de Greense functies van een gelaagde halfruimte.De Greense functies beschrijven de golfvoortplanting in de grond en wordenberekend met de directe stijfheidsmethode [113].

Cruciale parameters bij de voorspelling van grondtrillingen zijn de glijdings-modulus en de materiaaldemping van de grond. Deze karakteristieken wordenbepaald door laboratoriumproeven of in situ proeven. Laboratoriumproevenzijn meestal gebaseerd op de modale analyse van een proefstuk met beperkteafmetingen. In situ proeven zijn gebaseerd op de meting van de golfsnelheidtussen punten op grotere afstand. Deze proeven hebben een beperkte ruimtelijkeresolutie: ze laten niet toe de ruimtelijke variatie van de grondkarakteristieken opeen kleine schaal of op een grote diepte te bepalen. Het grondprofiel dat uit deproeven volgt is hierdoor niet uniek en dus onzeker. Als gevolg hiervan zijn ookde trillingsvoorspellingen onzeker.

Focus van de thesis

Het doel van deze thesis is de onzekerheid op de grondkarakteristieken te begrotenen de invloed van deze onzekerheid op de voorspelling van trillingen na te gaan.De focus ligt op de onzekerheid op de glijdingsmodulus wanneer die in situ bepaaldwordt door de spectrale analyse van oppervlaktegolven (E: Spectral Analysis ofSurface Waves, SASW). De methodologie is echter algemeen en kan ook toegepastworden op andere proeven of om de onzekerheid op de materiaaldemping tebepalen.

In de literatuur wordt een onderscheid gemaakt tussen aleatorische en episte-mische onzekerheid [62, 149, 227]. Aleatorische of inherente onzekerheid verwijstnaar een intrinsieke variabiliteit in een systeem. Dit type van onzekerheid isniet-reduceerbaar: ook wanneer alle informatie beschikbaar is, blijft de grootheidin kwestie onzeker. Typische voorbeelden zijn modeleigenschappen die bepaaldworden door productietoleranties of de winddruk op een gebouw. Epistemischeof subjectieve onzekerheid verwijst naar onzekerheid door het ontbreken vaninformatie. Dit type van onzekerheid is reduceerbaar: naarmate meer informatiebeschikbaar wordt, neemt de onzekerheid af en evolueert de onzekere grootheidnaar een deterministische waarde. De onzekerheid die in deze thesis beschouwdwordt, is van het epistemische type.

Alhoewel aleatorische en epistemische onzekerheid een compleet verschil-lende betekenis hebben, worden ze wiskundig op dezelfde wijze beschreven.Klassiek wordt een probabilistische aanpak gevolgd waarbij onzekere parametersgemodelleerd worden als stochastische variabelen of stochastische processen, diegekenmerkt worden door gemiddeldes, varianties, covarianties, kansverdelingen,

Samenvatting vii

en gemeenschappelijke kansverdelingen [119]. Het gebruik van stochastischeprocessen maakt het mogelijk de ruimtelijke variatie van onzekere parameters inrekening te brengen.

De reductie van epistemische onzekerheid door het uitvoeren van metingenwordt meestal begroot volgens een Bayesiaans schema [21]. Uitgaande van dereeds beschikbare informatie wordt een a priori stochastisch model opgesteld. Ditmodel wordt bijgewerkt aan de hand van de meetdata. Zo wordt een a posterioristochastisch model bekomen dat de a priori informatie en de meetdata combineert.

In de volgende sectie wordt een eenvoudig dynamisch grond-structuurinteractie-probleem besproken waarbij grondtrillingen gegenereerd worden door een hamer-impact op een kleine betonnen fundering. Dit probleem dient als toepassingsvoor-beeld doorheen de thesis. Het doel is de gemeten trillingen numeriek te simuleren,rekening houdend met de onzekerheid op de glijdingsmodulus van de grond.

In de daaropvolgende secties van dit hoofdstuk wordt uitgelegd hoe ditinteractieprobleem op deterministische en stochastische wijze wordt opgelost. Inbeide gevallen wordt de glijdingsmodulus van de grond bepaald aan de hand vaneen SASW-proef en wordt een subdomeinformulering voor dynamische grond-structuurinteractie gebruikt. In het stochastische geval wordt de glijdingsmodulusvan de grond gemodelleerd als een stochastisch proces waarvan de eigenschappenvolgens een Bayesiaans schema worden bepaald. Het stochastische dynamischegrond-structuurinteractieprobleem wordt opgelost met een Monte Carlomethode.

Een eenvoudig dynamisch grond-structuur-inter-

actieprobleem

In het kader van deze thesis is een experiment uitgevoerd op een site inLincent (Belgie) naast de hogesnelheidslijn L2 tussen Brussel en Luik [187].Grondtrillingen zijn gegenereerd door een hamerimpact op een betonnen funderingen geregistreerd door tien accelerometers in het vrije veld (figuur 1). Detransferfuncties H(ω) tussen de (verticale) kracht uitgeoefend door de hamer en deverticale verplaatsingen in het vrije veld zijn geschat. Figuur 2 toont de resultaten.De figuur geeft aan waar de coherentie Γ(ω) van de kracht en de verplaatsingde minimale waarde Γmin = 0.95 overschrijdt. In dit frequentiegebied, dat zichuitstrekt van 20 Hz tot 150 Hz op kleine afstanden en van 20 Hz tot 100 Hz opgrote afstanden van de fundering, is de signaal-ruisverhouding goed en kunnen demeetresultaten dienen als referentie voor de simulaties die in de volgende sectiesworden besproken.

Vergeleken met typische dynamische grond-structuurinteractieproblemen die devoorspelling van trillingen in de bebouwde omgeving beheersen is dit probleemrelatief eenvoudig: de fundering is niet ingebed in de grond, gedraagt zich als eenstar lichaam, en wordt geexciteerd met behulp van een hamer uitgerust met eenkrachtcel zodat de impactkracht rechtstreeks kan worden gemeten. Toch omvatdit probleem de belangrijkste elementen van meer complexe dynamische grond-

viii Samenvatting

Figuur 1: De meetopstelling op de site in Lincent.

structuurinteractieproblemen, in het bijzonder de transmissie van golven door degrond. De methodologie die in de volgende secties toegelicht wordt is bovendienalgemeen en dus ook toepasbaar in meer complexe situaties.

Deterministische voorspelling van trillingen

Deze sectie beschrijft de deterministische voorspelling van grondtrillingen met desubdomeinformulering ontwikkeld door Aubry en Clouteau [16, 36]. In een eerstesubsectie wordt de SASW-methode toegelicht. Met behulp van deze methodewordt de glijdingsmodulus van de grond op de site in Lincent bepaald. In eentweede subsectie wordt het resulterende grondprofiel gebruikt voor de voorspellingvan de trillingen gemeten op de site in Lincent. De resultaten worden vergelekenmet de experimentele data besproken in de vorige sectie.

Bepaling van de grondkarakteristieken

De SASW-methode is een niet-invasieve methode ter bepaling van de glijdings-modulus van ondiepe grondlagen [151, 231]. De methode wordt gebruikt omwegverhardingen te onderzoeken [152], om de kwaliteit van grondverbetering nate gaan [41], om de dikte van stortlagen te bepalen [114], en om de dynamischegrondkarakteristieken te identificeren voor de voorspelling van grondtrillingen[136, 143, 169].

Samenvatting ix

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32 m

Figuur 2: Modulus van de fundering-grondtransferfunctie H(ω) op (a) 4 m, (b)8 m, (c) 16 m, en (d) 32 m van het midden van de fundering. Een volle lijn geeftaan dat de coherentie Γ(ω) de waarde Γmin = 0.95 overschrijdt.

De SASW-methode is gebaseerd op het dispersieve karakter van oppervlakte-golven in een gelaagd medium. Bij lage frequenties zijn de golflengtes groot enreiken de oppervlaktegolven diep. Diepe grondlagen zijn meestal stijf, waardoorde fasesnelheid van de oppervlaktegolven hoog is. Naarmate de frequentie stijgt,wordt de golflengte kleiner en planten de oppervlaktegolven zich voort door minderdiepe lagen. Deze lagen zijn meestal slapper, waardoor de fasesnelheid daalt.

De SASW-methode bestaat uit drie stappen. In de eerste stap wordt eenexperiment uitgevoerd gelijkaardig aan het experiment in de vorige sectie:trillingen worden gegenereerd met behulp van een valgewicht, een impacthamer, ofeen hydraulische shaker en de respons in het vrije veld wordt gemeten met geofonenof accelerometers tot op een afstand van typisch 50 m. In de tweede stap wordtde experimentele dispersiecurve CE

R(ω) afgeleid van de fase van de transferfunctiestussen de signalen gemeten in het vrije veld. Hierbij wordt verondersteld datde respons op een voldoende grote afstand van de bron gedomineerd wordtdoor oppervlaktegolven. In de derde stap wordt een invers probleem opgelostter bepaling van de dynamische glijdingsmodulus van de grond. De directestijfheidsmethode [113] wordt gebruikt om de theoretische dispersiecurve CT

R (ω)te berekenen van een grond met een gegeven profiel. Dit profiel wordt iteratief

x Samenvatting

aangepast om de afstand tussen de theoretische en de experimentele dispersiecurvete minimaliseren.

In een gelaagde halfruimte bestaan meerdere oppervlaktegolven. In de SASW-methode wordt gewoonlijk verondersteld dat het golfveld aan het grondoppervlakgedomineerd wordt door de fundamentele oppervlaktegolf, dit is de oppervlaktegolfmet de laagste fasesnelheid. Deze veronderstelling is niet steeds correct: indiende grond bestaat uit stijve lagen bovenop een of meer slappe lagen kunnenoppervlaktegolven van hogere orde de verplaatsingen aan het grondoppervlakbeınvloeden [88, 216]. Daarom wordt in deze thesis de theoretische dispersiecurveCT

R (ω) niet gedefinieerd als de fasesnelheid van de fundamentele oppervlaktegolf,maar als de fasesnelheid waarbij de modulus van de Greense functie uG

zz in hetfrequentie-golfgetaldomein maximaal is [233]. De Greense functie uG

zz geeft deverticale verplaatsing ten gevolge van een verticale impactbelasting, beide aan hetgrondoppervlak. Deze aanpak resulteert in een effectieve dispersiecurve die bijiedere frequentie de fasesnelheid van de dominante oppervlaktegolf weergeeft.

In het kader van deze thesis is de SASW-methode toegepast op de site in Lincent,uitgaande van het experiment beschreven in de vorige sectie. De experimenteledispersiecurve CE

R(ω) is bepaald volgens de methode van Nazarian [151] op basisvan de fase van de transferfuncties tussen de versnellingen in het vrije veld.Figuur 3 toont het resultaat. Bij iedere frequentie wordt de fase van meerderetransferfuncties (tussen verschillende paren van ontvangers) beschouwd. Ditgeeft aanleiding tot verschillende schattingen van de experimentele dispersiecurveCE

R(ω). De experimentele dispersiecurve CER(ω) wordt gedefinieerd als de (vijfde

orde) polynomiale benadering van deze schattingen.

0 25 50 75 100 125 150 175 2000

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Figuur 3: Experimentele dispersiecurve (grijze punten) en polynomiale benadering(zwarte punten) voor de site in Lincent.

Vervolgens wordt een invers probleem opgelost ter bepaling van het grondprofielwaarvoor de afstand tussen theoretische dispersiecurve CT

R (ω) en de experimenteledispersiecurve CE

R(ω) minimaal is. Hiertoe wordt de volgende doelfunctie fobj

Samenvatting xi

geminimaliseerd:

fobj =

√

√

√

√

1

N

N∑

n=1

(

CTR (ωn) − CE

R(ωn))2

(1)

waarbij ωn een aantal frequenties zijn in het bereik waarbinnen de experimenteledispersiecurve CE

R(ω) is bepaald. De grond wordt gemodelleerd door middel vandrie homogene lagen op een homogene halfruimte. De lagen en de halfruimteworden gekarakteriseerd door een dikte d, een glijdingsmodulus µ, een coefficientvan Poisson ν, een dichtheid ρ, en een materiaaldempingsverhouding β. Decoefficient van Poisson ν, dichtheid ρ, en dempingsverhouding β zijn gekozenop basis van eerder uitgevoerde proeven op dezelfde site. De dikte d ende glijdingsmodulus µ van de verschillende lagen zijn de parameters in hetminimalisatieprobleem. Dit probleem wordt opgelost door middel van een lokaleoptimalisatiemethode. De resultaten zijn samengevat in tabel 1. Hierin zijnCs =

√

µ/ρ en Cp =√

(λ+ 2µ)/ρ de golfsnelheden van respectievelijk deschuifgolven en de drukgolven, met λ en µ de coefficienten van Lame.

Laag d Cs Cp β ρ[m] [m/s] [m/s] [−] [kg/m3]

1 0.57 140 280 0.03 18002 1.44 154 308 0.03 18003 3.43 220 440 0.03 18004 ∞ 287 574 0.03 1800

Tabel 1: Grondprofiel afgeleid uit de SASW-proef.

Figuur 4 vergelijkt de theoretische dispersiecurve CTR(ω) van het geıdentificeerde

grondprofiel met de experimentele curve CER(ω). De overeenkomst is uitstekend.

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Figuur 4: Experimentele (zwarte punten) en theoretische (grijze lijn)dispersiecurve voor de site in Lincent.

xii Samenvatting

De fundering-grondtransferfuncties

In deze subsectie wordt het experiment in Lincent gesimuleerd. De fundering-grondtransferfuncties H(ω) worden voorspeld op basis van het grondprofiel afgeleidin de vorige subsectie en vervolgens vergeleken met de experimentele databesproken in de vorige sectie.

De voorspelling van de transferfuncties H(ω) vereist de oplossing van eendynamisch grond-structuurinteractieprobleem. Dit gebeurt met een methodegebaseerd op de subdomeinformulering ontwikkeld door Aubry en Clouteau[16, 36].

De subdomeinformulering is gebaseerd op de projectie van het verplaatsingsveldop een kinematische basis bestaande uit een beperkt aantal basisvectoren. Debasisvectoren komen overeen met de eigenmodes ψbm van de structuur met vrijerandvoorwaarden en de overeenkomstige afgestraalde golfvelden ψsm in de grond.De modale coordinaten αm volgen uit het evenwicht van de structuur:

[

Kb − ω2Mb + Ks

]

α = fb (2)

met Kb en Mb de modale stijfheids- en massamatrices van de structuur, Ks

de modale impedantiematrix van de grond, en fb de modale belasting op destructuur. De matrices Kb en Mb worden geassembleerd met de eindige-elementenmethode en de matrix Ks met de randelementenmethode [25, 54]. Derandelementenmethode is gebaseerd op de Greense functies uG

ij(x′,x, ω) van de

grond. Deze functies geven de verplaatsingen in de richting ej in het punt x

ten gevolge van een harmonische puntlast die aangrijpt in de richting ei in hetpunt x′. De Greense functies worden berekend in het frequentie-golfgetaldomeinmet de directe stijfheidsmethode [113] en getransformeerd naar het ruimtelijkedomein door middel van een inverse Hankeltransformatie. Deze transformatiewordt numeriek geevalueerd door middel van een methode ontwikkeld door Talman[212] die efficient gebruik maakt van het FFT-algoritme. Deze methode vereist datde Greense functies logaritmisch bemonsterd worden in het golfgetaldomein en hetruimtelijke domein. Een logaritmische bemonstering is voordelig bij problemenwaarin sterk verschillende ruimtelijke schalen een rol spelen, zoals dynamischegrond-structuurinteractieproblemen, waar zowel de interactie tussen nabijgelegenpunten op het grensvlak tussen grond en structuur als de afstraling van golvennaar verafgelegen punten in het vrije veld van belang zijn.

De subdomeinformulering is gebruikt om de fundering-grondtransferfunctiesH(ω) gemeten in Lincent te voorspellen. De grond is gemodelleerd zoalsaangegeven in tabel 1 en de fundering is gemodelleerd als een star lichaam datenkel een verticale verplaatsing ondergaat. De resultaten worden weergegevenin figuur 5, samen met de experimentele data beschreven in de vorige sectie.In het frequentiebereik waar de coherentie van de experimentele data goedis, is de overeenkomst tussen de voorspelde en gemeten transferfuncties H(ω)aanvaardbaar tot 50 Hz. Daarboven worden de experimentele transferfunctiesoverschat. Deze overschatting is niet eenvoudig te verklaren. Mogelijk is het

Samenvatting xiii

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32 m

Figuur 5: Modulus van de gemeten (zwarte lijn) en voorspelde (grijze lijn)fundering-grondtransferfunctie H(ω) op (a) 4 m, (b) 8 m, (c) 16 m, en (d) 32 m vanhet midden van de fundering. Een volle zwarte lijn geeft aan dat de coherentieΓ(ω) de waarde Γmin = 0.95 overschrijdt.

grondprofiel dat volgt uit de SASW-proef niet correct. De robuustheid van deinversieprocedure die gebruikt is ter bepaling van dit profiel wordt in de volgendesectie onderzocht.

Stochastische voorspelling van trillingen

Deze sectie beschrijft de stochastische voorspelling van grondtrillingen door middelvan Monte Carlosimulaties. De Monte Carlosimulaties omvatten het oplossenvan een groot aantal willekeurig gegenereerde deterministische dynamische grond-structuurinteractieproblemen. In een eerste subsectie wordt aangegeven hoewillekeurige grondprofielen gegenereerd worden waarvoor de theoretische dis-persiecurve overeenkomt met de experimentele curve van de site in Lincent.In een volgende subsectie worden deze profielen gebruikt om de fundering-grondtransferfuncties te voorspellen. De resultaten worden vergeleken met deexperimentele fundering-grondtransferfuncties.

xiv Samenvatting

Bepaling van de grondkarakteristieken

In deze subsectie wordt het inverse probleem in de SASW-methode benaderdvanuit een probabilistische invalshoek. Het doel is een ensemble van grond-profielen te bepalen waarvoor de theoretische dispersiecurve overeenkomt metde experimentele curve van de site in Lincent. Dit ensemble wordt gebruiktom de resolutie van de SASW-methode te beoordelen en om grondtrillingen tevoorspellen, rekening houdend met de onzekerheid op de grondkarakteristieken.

Het stochastische inverse probleem wordt opgelost volgens een Bayesiaansschema [21]. Eerst wordt een a priori stochastisch model opgesteld op basisvan de informatie over de grondkarakteristieken die beschikbaar is voor deuitvoering van de SASW-proef. Dit model kent aan elk denkbaar grondprofieleen a priori waarschijnlijkheid toe. Daarna worden de resultaten van de SASW-proef gebruikt om voor elk grondprofiel de aannemelijkheid (E: likelihood)te definieren. De aannemelijkheid geeft aan in welke mate de theoretischedispersiecurve overeenkomt met de experimentele. Tenslotte wordt het a posterioristochastische model bepaald: aan elk grondprofiel wordt een a posterioriwaarschijnlijkheid toegekend gelijk aan het (genormaliseerde) product van de apriori waarschijnlijkheid en de aannemelijkheid. Het a posteriori model is dus eenweergave van de vooraf beschikbare informatie gecombineerd met de resultatenvan de SASW-proef.

De grond wordt gemodelleerd door middel van een laag met dikte L, waarin deglijdingsmodulus µ varieert met de diepte, bovenop een homogene halfruimte. Inde laag wordt de glijdingsmodulus gemodelleerd als een stochastisch proces µ(z, θ),met z de verticale coordinaat en θ de coordinaat in de stochastische dimensie[119]. In de onderliggende halfruimte is de glijdingsmodulus gelijk aan µ(L, θ). Decoefficient van Poisson ν, de materiaaldempingsverhouding β, en de dichtheid ρworden gekend verondersteld en zowel in de laag als in de halfruimte gelijkgesteldaan ν = 0.33, β = 0.03, en ρ = 1800 kg/m3.

In het a priori model wordt het stochastische proces µ(z, θ) stationair veronder-steld en gekarakteriseerd door een marginale kansdichtheidsfunctie (E: ProbabilityDensity Function, PDF) pµ(µ) en een covariantiefunctie Cµ(z). De marginale PDFpµ(µ) beschrijft de verdeling van de glijdingsmodulus op een welbepaald punt.De covariantiefunctie Cµ(z) beschrijft de variatie van de glijdingsmodulus met dediepte.

De a priori marginale PDF pµ(µ) en covariantiefunctie Cµ(z) zijn weergegevenin figuur 6. De marginale PDF pµ(µ) is uniform verdeeld tussen 10.1 MPaen 162 MPa. De overeenkomstige waarden van de schuifgolfsnelheid Cs zijn75 m/s en 300 m/s. Dit zijn typische waarden voor ondiepe grondlagen inBelgie. De functie Cµ(z) is een Materncovariantiefunctie [92, 95, 197] met eencorrelatielengte lc = 0.25 m. De correlatielengte lc is een maat voor de schaal vande ruimtelijke variatie van het stochastische proces µ(z, θ). Door een relatief kleinecorrelatielengte te kiezen, zal de a priori glijdingsmodulus scherpe variaties metde diepte vertonen. Deze variaties kunnen vergeleken worden met die van de a

Samenvatting xv

(a)0 50 100 150 200

0

2

4

6

8x 10

−3

Shear modulus [MPa]

Pro

babi

lity

dens

ity [M

Pa−

1 ]

(b)0 0.5 1 1.5 2

0

500

1000

1500

2000

Distance [m]

Cov

aria

nce

[MP

a2 ]Figuur 6: A priori (a) marginale PDF pµ(µ) en (b) covariantiefunctie Cµ(z) vande glijdingsmodulus µ(z, θ).

posteriori glijdingsmodulus om de resolutie van de SASW-methode te bepalen.De stochastische glijdingsmodulus µ(z, θ) wordt gemodelleerd als een niet-

Gaussiaans translatieproces [83]:

µ(z, θ) = F−1µ

(

FG

(

η(x, θ))

)

(3)

Hierin is η(x, θ) een standaard-Gaussiaans proces en zijn Fµ(µ) en FG(η) demarginale cumulatieve verdelingsfuncties (E: Cumulative Distribution Function,CDF) van de processen µ(z, θ) en η(z, θ). De covariantiefunctie Cη(z) vanhet Gaussiaanse proces η(z, θ) wordt zo bepaald dat vergelijking (3) resulteertin een proces µ(z, θ) met de vooropgestelde covariantiefunctie Cµ(z). HetGaussiaanse proces η(z, θ) wordt gediscretiseerd met behulp van de Karhunen-Loevedecompositie [78]:

η(z, θ) =

M∑

k=1

√

λkfk(z)ξk(θ) (4)

Hierin zijn λk en fk(z) de M grootste eigenwaarden en overeenkomstigeeigenfuncties van de covariantiefunctie Cη(z). De parameters ξk(θ) zijn onderlingonafhankelijke standaard-Gaussiaanse variabelen. De variabelen ξk(θ) bepalenhet stochastische verloop het Gaussiaanse proces η(z, θ) en dus ook van deglijdingsmodulus µ(z, θ). Het a priori grondmodel kan dus gekarakteriseerd wordendoor een a priori PDF ρξ(ξ) in termen van de variabelen ξk(θ):

ρξ(ξ) =1

(2π)M/2exp

(

−‖ξ‖2

2

)

(5)

Hierin is ξ(θ) een stochastische vector die de variabelen ξk(θ) groepeert.In een Monte Carlosimulatie worden willekeurige realisaties van de variabelen

ξk(θ) geıntroduceerd in vergelijking (4). Zo wordt een realisatie van het proces

xvi Samenvatting

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [MPa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

0

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figuur 7: (a) Tien realisaties van de a priori glijdingsmodulus µ(z, θ) en (b)overeenkomstige theoretische dispersiecurves CT

R (ω) (grijze lijnen), vergeleken metde experimentele dispersiecurve CE

R(ω) (zwarte punten).

η(z, θ) bekomen die vervolgens vertaald wordt in een realisatie van de stochastischeglijdingsmodulus µ(z, θ) met behulp van vergelijking (3). Figuur 7 toont eenaantal realisaties van de a priori glijdingsmodulus µ(z, θ) en de overeenkomstigetheoretische dispersiecurves CT

R (ω). De theoretische dispersiecurves CTR(ω) komen

niet overeen met de experimentele curve CER(ω) doordat het a priori model geen

rekening houdt met de experimentele data.Voor elk profiel in het a priori model wordt de aannemelijkheidsfunctie Lξ(ξ)

gedefinieerd als:

Lξ(ξ) =

0 als maxω

∣

∣CTR(ω) − CE

R(ω)∣

∣ > ∆CR

12∆CR

als maxω

∣

∣CTR(ω) − CE

R(ω)∣

∣ ≤ ∆CR

(6)

met ∆CR = 5 m/s. Alle profielen waarvoor de afwijking van de theoretischedispersiecurve CT

R(ω) ten opzichte van de experimentele curve CTR (ω) kleiner is

dan de drempelwaarde ∆CR zijn aanvaardbaar en even aannemelijk. Alle andereprofielen zijn onaanvaardbaar.

De a posteriori PDF σξ(ξ) is gedefinieerd als:

σξ(ξ) = kρξ(ξ)Lξ(ξ) (7)

waarin de normalisatieconstante k zo gekozen wordt dat de integraal van de PDFσξ(ξ) gelijk is aan 1. Vergelijking (7) laat niet toe de a posteriori PDF σξ(ξ)expliciet te bepalen doordat er geen gesloten uitdrukking beschikbaar is voor deaannemelijkheidsfunctie Lξ(ξ). De vergelijking laat wel toe een populatie vangrondprofielen te bekomen die verdeeld is volgens de a posteriori PDF σξ(ξ).Hiertoe wordt een Markovketen-Monte Carlomethode toegepast. Met behulp vanhet Metropolis-Hastingsalgoritme [98, 148] wordt een Markovketen geconstrueerdin de vectorruimte van de stochastische variabelen ξ(θ). De keten bestaat

Samenvatting xvii

uit opeenvolgende toestanden ξi die elk overeenkomen met een aanvaardbaargrondprofiel. Een stap in het Metropolis-Hastingsalgoritme verloopt als volgt.Uitgaande van de huidige toestand ξi wordt op willekeurige wijze een kandidaatξ′i+1 voor de volgende toestand ξi+1 gegenereerd. Dit gebeurt met behulp vaneen conditionele PDF q(ξ′i+1|ξi) die afhankelijk is van de huidige toestand ξi. Dekandidaat ξ′i+1 wordt geaccepteerd met de volgende waarschijnlijkheid:

r(ξi, ξ′i+1) = min

σξ(ξ′i+1)

σξ(ξi)

q(ξi|ξ′i+1)

q(ξ′i+1|ξi), 1

(8)

Als de kandidaat niet geaccepteerd wordt, wordt de volgende toestand ξi+1 gelijkgesteld aan de huidige toestand ξi.

In deze studie is de Markovketen gestart vanuit een toestand ξ1 met aanneme-lijkheid Lξ(ξ1) = 1 en gestopt na 106 stappen. De conditionele PDF q(ξ′i+1|ξi) isgedefinieerd als een Gaussiaanse PDF gecentreerd rond de huidige toestand ξi:

q(ξ′i+1|ξi) =1

(2π)M/2σMqexp

(

−∥

∥ξ′i+1 − ξi∥

∥

2

2σ2Mq

)

(9)

waarbij de standaardafwijking σq = 0.08 de stapgrootte bepaalt.Figuur 8a toont een aantal realisaties van de a posteriori glijdingsmodulus. Deze

realisaties komen overeen met toestanden in de Markovketen. De overeenkomstigetheoretische dispersiecurves worden getoond in figuur 8b. Deze curves komengoed overeen met de experimentele curve. Alle grondprofielen in figuur 8a zijn dusaanvaardbaar.

Vergeleken met de a priori glijdingsmodulus (figuur 7a) is de variabiliteit vande a posteriori glijdingsmodulus (figuur 8a) kleiner, in het bijzonder in eenzone dichtbij het grondoppervlak. De grootschalige variaties van de a posterioriglijdingsmodulus zijn minder uitgesproken die van de a priori glijdingsmodulus,terwijl de kleinschalige variaties in beide gevallen vergelijkbaar zijn. Dit wijsterop dat de resolutie van de SASW-proef beperkt is in termen van diepte enruimtelijke schaal. Grootschalige variaties van de glijdingsmodulus dicht bij hetgrondoppervlak worden goed geıdentificeerd. Op kleinere schaal en op groterediepte is de resolutie minder goed.

De fundering-grondtransferfuncties

In deze subsectie wordt het experiment in Lincent opnieuw gesimuleerd, uitgaandevan het stochastische grondmodel bepaald in de vorige subsectie. De fundering-grondtransferfuncties H(ω) worden voorspeld door middel van een Monte Carlo-simulatie voor 1000 realisaties van de glijdingsmodulus µ(z, θ). Voor elke realisatiewordt een deterministisch dynamisch grond-structuurinteractieprobleem opgelostdoor middel van de subdomeinformulering toegelicht in de vorige sectie. De MonteCarlosimulatie is uitgevoerd aan de hand van zowel het a priori als het a posterioristochastische model.

xviii Samenvatting

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [MPa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

0

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figuur 8: (a) Tien realisaties van de a posteriori glijdingsmodulus µ(z, θ) en (b)overeenkomstige theoretische dispersiecurves CT

R (ω) (grijze lijnen), vergeleken metde experimentele dispersiecurve CE

R(ω) (zwarte punten).

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figuur 9: Tien realisaties (grijze lijnen) en 95 % betrouwbaarheidsinterval (grijzezone) van de modulus van de a priori transferfunctie H(ω) vergeleken met deexperimentele data (zwarte lijn) op (a) 4 m, (b) 8 m, (c) 16 m, en (d) 32 m van hetmidden van de fundering. Een volle zwarte lijn geeft aan dat de coherentie Γ(ω)de waarde Γmin = 0.95 overschrijdt.

Samenvatting xix

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figuur 10: Tien realisaties (grijze lijnen) en 95 % betrouwbaarheidsinterval (grijzezone) van de modulus van de a posteriori transferfunctie H(ω) vergeleken met deexperimentele data (zwarte lijn) op (a) 4 m, (b) 8 m, (c) 16 m, en (d) 32 m van hetmidden van de fundering. Een volle zwarte lijn geeft aan dat de coherentie Γ(ω)de waarde Γmin = 0.95 overschrijdt.

Figuren 9 en 10 tonen tien realisaties en het 95 % betrouwbaarheidsinterval vande modulus van respectievelijk de a priori en de a posteriori transferfuncties H(ω).Vergeleken met de a priori transferfuncties is de variabiliteit van de a posterioritransferfuncties duidelijk kleiner. De reductie van de variabiliteit is het grootstin het frequentiebereik van 20 Hz tot 50 Hz. In dit bereik planten de golven in degrond zich voort door relatief ondiepe lagen maar worden ze niet beınvloed doorkleinschalige variaties van de grondkarakteristieken. De resolutie van de SASW-proef laat in dit frequentiebereik robuuste voorspellingen van grondtrillingen toe(op voorwaarde dat de materiaaldemping van de grond gekend is).

Bij lagere frequenties reiken de golven dieper. De glijdingsmodulus van dieperegrondlagen kan niet eenduidig bepaald worden door middel van de SASW-proef.In dit frequentiegebied wordt dus een grote variabiliteit van de a posterioritransferfuncties verwacht. Dit echter is niet zichtbaar in figuur 10 door de beperktedikte L van de laag waarbinnen de variatie van de glijdingsmodulus µ(z, θ) inrekening is gebracht.

Bij hogere frequenties wordt de dominante golflengte in de grond kleiner en

xx Samenvatting

worden de golven beınvloed door de kleinschalige variaties van de glijdingsmodulusµ(z, θ). Deze kleinschalige variaties kunnen niet eenduidig bepaald worden doormiddel van de SASW-proef. De variabiliteit van de a posteriori transferfuncties isbijgevolg relatief groot: de SASW-proef laat in dit frequentiegebied geen robuustetrillingsvoorspellingen toe. De robuustheid van de voorspellingen kan verbeterdworden door ter bepaling van de glijdingsmodulus een techniek te gebruiken meteen fijnere ruimtelijke resolutie.

De a posteriori transferfuncties H(ω) komen bij lage frequenties goed overeenmet de experimentele data (figuur 10). Bij hoge frequenties is nog steeds eenoverschatting van de experimentele transferfuncties zichtbaar. Deze overschattingis moeilijk te verklaren. Mogelijke oorzaken zijn onder meer de aanname datde grond horizontaal gelaagd is, het feit dat de materiaaldemping constantverondersteld wordt (in ruimtelijke en stochastische dimensie), en het constitutiefgedrag van de grond net onder de fundering dat mogelijk niet-lineair is.

Originele bijdragen van de thesis

De berekening van de tweedimensionale en driedimensionale Greense verplaatsings-en spanningsfuncties van een gelaagde grond wordt in detail behandeld. Alleelementen van de berekening behoren al tot de state of the art sinds 25 jaar,maar in deze thesis zijn ze samengebracht in een coherent en volledig overzicht.Dit overzicht is gebaseerd op een cursustekst over seismische golfvoortplanting [46]waarin de berekening van de tweedimensionale Greense verplaatsingsfuncties wordtbesproken. In het kader van deze thesis is een MATLAB-toolbox (ElastoDynamicToolbox 2.0, EDT 2.0) ontwikkeld voor seismische golfvoortplanting. Veelaandacht is besteed aan een efficiente implementatie aangezien deze toolboxgebruikt wordt in Monte Carlosimulaties voor de berekening van grote aantallenGreense functies.

De Greense functies worden berekend in het golfgetaldomein en getransformeerdnaar het ruimtelijke domein door middel van inverse Fourier- en Hankeltransfor-maties. Hiertoe wordt een algoritme ontwikkeld door Talman [212] aangewend.Dit algoritme vereist dat de Greense functies logaritmisch bemonsterd wordenin het golfgetaldomein en het ruimtelijke domein. Daardoor is dit algoritmebijzonder geschikt voor problemen waarbij sterk verschillende ruimtelijke schaleneen rol spelen, zoals dynamische grond-structuurinteractieproblemen, waar zowelde interactie tussen nabijgelegen punten op het grensvlak tussen grond en structuurals de afstraling van golven naar verafgelegen punten in het vrije veld van belangzijn. Het originele algoritme is verbeterd door het gebruik van een window eneen filter om artefacten ten gevolge van het Gibbseffect te onderdrukken. Verderzijn richtlijnen geformuleerd voor de bemonstering van de Greense functies in hetgolfgetaldomein. Het algoritme is geıntegreerd in de MATLAB-toolbox EDT 2.0.

Componenten van de stochastische eindige-elementenmethode [78] zijn gecom-bineerd met de directe stijfheidsmethode [113] voor de modellering van golfvoort-

Samenvatting xxi

planting in grond waarvan de glijdingsmodulus stochastisch varieert met dediepte. Deze aanpak maakt het mogelijk rekening te houden met de onzekerheidop de grondkarakteristieken in een onbegrensd domein. In de literatuur overstochastische mechanica worden moduli van Young en glijdingsmoduli dikwijlsgemodelleerd als Gaussiaanse variabelen of processen [32, 78, 82, 109, 140, 211,229]. De aanname van een Gaussiaanse verdeling is echter fysisch niet realistischdoor de eindige kans dat een Gaussiaanse variabele een negatieve waarde aanneemt.In deze thesis wordt de glijdingsmodulus daarom gemodelleerd als een niet-Gaussiaans translatieproces [83].

Het inverse probleem in de SASW-methode wordt opgelost volgens eenBayesiaans schema. Het resulterende stochastische grondmodel wordt gebruiktom de onzekerheid in de voorspelling van grondtrillingen te beoordelen. Devoorspellingen worden vergeleken met experimentele resultaten. Om de con-vergentiesnelheid van de inversieprocedure te maximaliseren, worden enkel decomponenten van de glijdingsmodulus die een invloed hebben op de voorspeldegrondtrillingen geıdentificeerd.

Suggesties voor verder onderzoek

De keuze van het a priori stochastische model bij de Bayesiaanse bepalingvan de grondkarakteristieken is niet eenvoudig. Vooral de keuze van dea priori correlatielengte is delicaat. Deze lengte bepaalt de schaal van deruimtelijke variatie van de a priori glijdingsmodulus, en dus ook de a posterioriglijdingsmodulus. Vanuit Bayesiaans standpunt geeft de correlatielengte aan inwelke mate men ervan overtuigd is dat de glijdingsmodulus scherpe variatiesvertoont. De keuze van de a priori correlatielengte is dus per definitie subjectief.Toch kan verder onderzoek gericht zijn op een beter onderbouwde keuze van de apriori correlatielengte.

Hoewel enkel de SASW-proef behandeld wordt in deze thesis, is de methodologiealgemeen en kan ze ook toegepast worden bij de bepaling van de ruimtelijkeresolutie van andere proeven, zoals de seismische sondering (E: Seismic ConePenetration Test, SCPT). Verder laat de methodologie toe proeven te ontwerpenwaarvan de resolutie volstaat voor robuuste trillingsvoorspellingen in specifiekesituaties. Zo kan de afstand tussen de geofonen die gebruikt worden in de SCPT-proef geoptimaliseerd worden zodanig dat de onzekerheid in de voorspelling vantrillingen door treinverkeer minimaal wordt.

Dezelfde methodologie is toepasbaar bij de bepaling van andere dynamischegrondkarakteristieken, zoals de materiaaldemping. De materiaaldemping heeft eengrote invloed op de transmissie van golven door de grond. Voor een realistischevoorspelling van grondtrillingen is de bepaling van de materiaaldemping bijgevolgessentieel. Het belangrijkste obstakel is het ontbreken van een gevestigde proefom de materiaaldemping te bepalen, zoals de SASW-proef en de SCPT-proefvoor de bepaling van de glijdingsmodulus. Verder onderzoek kan daarom gericht

xxii Samenvatting

zijn op de ontwikkeling of verbetering van nieuwe of bestaande technieken om demateriaaldemping van grond te bepalen.

In dit werk wordt de variatie van de grondkarakteristieken in horizontale richtinggenegeerd. In werkelijkheid is deze variatie vermoedelijk minder belangrijk dan devariatie in verticale richting, maar heeft ze zeker een invloed op de transmissievan golven door de grond. De methodologie uitgewerkt in deze thesis kanaangewend worden om de horizontale variatie van de grondkarakteristieken teidentificeren, op voorwaarde dat er een voorwaarts model beschikbaar is waarmeede dispersiecurve van een volledig inhomogene grond kan worden berekend. Dedirecte stijfheidsmethode is hiervoor niet bruikbaar. De eindige-elementenmethodevormt een alternatief, maar vergt veel rekenkracht. Methodes gebaseerd opde verrijking van de polynomiale benadering van het golfveld in de eindige-elementenmethode kunnen een oplossing bieden [120, 232].

In deze thesis worden grondtrillingen beschouwd ten gevolge van een hamer-impact op een kleine fundering. Verder onderzoek kan gericht zijn op de invloedvan onzekere grondkarakteristieken op trillingen afkomstig van andere bronnen,zoals weg- en treinverkeer. Door het gebruik van een Monte Carlomethode zijnde bestaande deterministische bronmodellen [135, 136] direct toepasbaar. Hetgebruik van complexe bronmodellen in Monte Carlosimulaties vergt echter veelrekenkracht. De rekentijd van de bestaande bronmodellen voor verkeerstrillingenwordt bepaald door inverse Fouriertransformaties van het golfgetaldomein naar hetruimtelijke domein. Deze transformaties worden geevalueerd door middel van eenFilonmethode. Het gebruik van het logaritmische Fouriertransformatie-algoritmebeschreven in deze thesis kan leiden tot een efficienter bronmodel dat bruikbaar isin Monte Carlosimulaties.

Tenslotte kan het stochastische grondmodel voorgesteld in deze thesis gekoppeldworden aan een stochastisch gebouwmodel om alle onzekerheden in rekening tebrengen die een invloed hebben op de voorspelling van trillingen en herafgestraaldgeluid in gebouwen. De onzekerheden in het gebouwmodel zijn afkomstigvan een groot aantal parameters. Ze zijn zeer moeilijk te beschrijven op deklassieke, parametrische wijze waarbij alle onzekere parameters gemodelleerdworden als stochastische variabelen of processen. Methodes zoals statistischeenergieanalyse (E: Statistical Energy Analysis, SEA) vormen een alternatiefvoor trillingsvoorspellingen bij hoge frequenties, maar zijn niet bruikbaar inhet frequentiegebied dat hier van belang is. Een andere optie is de niet-parametrische stochastische methode [200], waarbij de onzekerheid in het model opeen globale manier gekenmerkt wordt door een klein aantal dispersieparameters.Deze methode is reeds gebruikt voor de voorspelling van trillingen in gebouwen[12]. Daarbij is rekening gehouden met de onzekerheden in het gebouwmodel,maar de grond is gemodelleerd als deterministisch. Het stochastische grondmodelvoorgesteld in deze thesis laat toe ook de onzekerheid op de grondkarakteristiekenin rekening te brengen.

Samenvatting xxiii

Organisatie van de tekst

Deterministisch model Stochastisch model

Theo

retisc

he

ach

terg

rond

Tri

llin

gsv

oors

pel

lingen

Bep

aling

van

de

gro

ndka

rakte

rist

ieken

2. De directestijfheidsmethode voorgelaagde gronden

3. Greense functies vangelaagde gronden

4. Dynamischegrond-structuurinteractie

5. De SASW-methode

6. Stochastische mechanica

7. Golfvoortplanting ingrond met onzekerekarakteristieken

8. Stochastische bepalingvan de grond-karakteristieken met deSASW-methode

Figuur 11: Organisatie van de tekst.

De thesis bestaat uit twee delen waarin de deterministische en de stochastischevoorspelling van grondtrillingen behandeld worden (figuur 11). Het stochastischeprobleem wordt opgelost met een Monte Carlomethode. Daarbij wordt een grootaantal deterministische problemen opgelost. Veel aandacht is daarom besteed aande efficiente oplossing van het deterministische probleem, dat in detail beschrevenwordt in deze tekst.

Beide delen beginnen met een of twee hoofdstukken over de theoretischeachtergrond van het probleem, gevolgd door een hoofdstuk waarin het voorwaartseprobleem van de voorspelling van trillingen behandeld wordt (op basis van eensynthetisch grondprofiel), en een hoofdstuk over het inverse probleem in de SASW-methode (die aangewend wordt om het grondprofiel in het voorwaartse probleembij te werken).

xxiv Samenvatting

Hoofdstuk 1 bevat de inleiding van de thesis. Het onderwerp wordt gesitueerd,de eigen bijdragen worden aangegeven, en de organisatie van de tekst wordttoegelicht. Daarnaast wordt een experiment beschreven dat uitgevoerd is inLincent, waarbij grondtrillingen gegenereerd zijn door een hamerimpact op eenkleine fundering. Dit dynamisch grond-structuurinteractieprobleem dient alstoepassingsvoorbeeld doorheen de thesis.

Hoofdstuk 2 geeft een overzicht van de directe stijfheidsmethode voor gelaagdemedia [113]. De directe stijfheidsmethode wordt gebruikt om de Greense functies(in het golfgetaldomein) en de theoretische dispersiecurve van grond te berekenen.

Hoofdstuk 3 behandelt de transformatie van de tweedimensionale en dedriedimensionale Greense functies van het golfgetaldomein naar het ruimtelijkedomein. Analytische uitdrukkingen worden afgeleid en een efficient algoritme voorde transformaties wordt beschreven.

Hoofdstuk 4 geeft een kort overzicht van de subdomeinformulering voordynamische grond-structuurinteractie ontwikkeld door Aubry en Clouteau [16,36]. Deze methode wordt toegepast om de fundering-grondtransferfunctiesgemeten in Lincent te simuleren. De grond wordt gemodelleerd met eenrandelementenformulering [25, 54], die gebaseerd is op de driedimensionale Greensefuncties van een gelaagde halfruimte, besproken in hoofstuk 3. De simulaties zijngebaseerd op een synthetisch grondprofiel.

Hoofdstuk 5 beschrijft de SASW-methode [151, 231]. De berekening van deexperimentele en de theoretische dispersiecurve wordt toegelicht. De methodewordt toegepast ter bepaling van de glijdingsmodulus van de grond in Lincent. Opbasis van het resulterende grondprofiel worden de fundering-grondtransferfunctiesherberekend en vervolgens vergeleken met de experimentele data.

Hoofdstuk 6 schetst de waarschijnlijkheidstheorie van Kolmogorov, die dientals theoretisch raamwerk waarbinnen stochastische vectoren en processen wordengedefinieerd. Methodes om stochastische vectoren en processen te modellerenen te simuleren worden besproken. Deze methodes worden gebruikt in destochastische eindige-elementenmethode [78]. Drie frequent voorkomende vor-men van de stochastische eindige-elementenmethode worden beschouwd in dithoofdstuk: de perturbatie-stochastische eindige-elementenmethode, de spectralestochastische eindige-elementenmethode, en de Monte Carlo stochastische eindige-elementenmethode.

Hoofdstuk 7 behandelt golfvoortplanting in gronden met onzekere eigen-schappen. Componenten van de stochastische eindige-elementenmethode wor-den toegepast op het dynamische grond-structuurinteractieprobleem bespro-

Samenvatting xxv

ken in hoofdstuk 4. Eerst wordt de invloed van een onzekere glijdings-modulus, coefficient van Poisson, dichtheid, en materiaaldemping op de fundering-grondtransferfuncties nagegaan. Daarbij wordt aangenomen dat deze eigenschap-pen niet varieren in de ruimte. Vervolgens wordt het effect van een onzekerevariatie van de glijdingsmodulus met de diepte onderzocht. De simulaties in dithoofdstuk zijn gebaseerd op een synthetisch stochastisch grondmodel.

Hoofdstuk 8 benadert het inverse probleem in de SASW-methode op eenprobabilistische manier om te komen tot een stochastisch grondmodel dat rekeninghoudt met de onzekerheid in de SASW-methode. Een Bayesiaanse werkwijze wordtgevolgd waarbij a priori informatie gecombineerd wordt met de gegevens afkomstigvan een SASW-proef. Het resulterende a posteriori stochastische grondmodelwordt vervolgens gebruikt om de fundering-grondtransferfuncties uit hoofdstuk 7te herberekenen. De resultaten worden tenslotte vergeleken met de experimenteledata gemeten in Lincent.

Hoofdstuk 9 vat de conclusies van de thesis samen en geeft suggesties voorverder onderzoek.

De originele bijdragen van de thesis zijn voornamelijk gesitueerd in de hoofd-stukken 3, 7, en 8. De andere hoofdstukken geven de huidige state of the art weeren zijn opgenomen in de thesis om een coherent raamwerk te scheppen voor deontwikkeling van de methodologie.

xxvi Samenvatting

Summary

In recent years, several numerical models have been developed to predict ground-borne vibrations in the built environment. These models contain elements relatedto the characterization of the source, the transmission of waves through the soil,and the response of the building. The transmission of waves through the soilis governed by the dynamic soil properties. Crucial properties are the dynamicshear modulus and the material damping ratio. These properties are determinedby means of in situ tests or laboratory tests. The complete characterizationof the spatial variation of the soil properties is practically impossible, however.As a result, soil models used in ground vibration predictions are subjected touncertainty.

In the present thesis, the impact of this uncertainty on the prediction of groundvibrations is studied. The focus is on the dynamic shear modulus. A probabilisticapproach is followed, using a random process to model the variation of the shearmodulus with depth. The variation in the horizontal direction is neglected.

The uncertain shear modulus is determined from a Spectral Analysis of SurfaceWaves (SASW) test, following a Bayesian approach. The prior information on theshear modulus, which is available before the SASW test is performed, is combinedwith the experimental data from the SASW test to obtain a posterior stochasticsoil model. The posterior stochastic soil model is sampled by means of a Markovchain Monte Carlo method. This leads to an ensemble of soil profiles that fit theexperimental data. The resolution of the SASW test is assessed through inspectionof this ensemble.

Next, the prediction of ground vibrations is considered. The free field vibrationsdue to a hammer impact on a small foundation are predicted. A Monte Carlomethod is applied, using the ensemble of soil profiles determined from the SASWtest. The results are compared with experimental data. It is observed that thevariability of the free field response is frequency dependent. In the low frequencyrange, the resolution of the SASW test is sufficient and the variability of theresponse is limited. At higher frequencies, the uncertainty involved in the SASWtest affects the predicted response and its variability increases.

xxvii

xxviii Summary

List of Symbols

The following list provides an overview of symbols used throughout the text. Thephysical meaning of the symbols is explained in the text. Vectors, matrices andtensors are denoted by bold characters. In the text, variables in the frequency-space domain are indicated by a hat, and variables in the frequency-wavenumberdomain are indicated by a tilde. No hats or tildes are used in the following list,however.

The general symbols and conventions and the abbreviations are collected in thefirst two sections. The other symbols are categorized in sections referring to thechapters where they are first introduced.

General symbols and conventions

(x, y, z) Cartesian coordinates(r, θ, z) cylindrical coordinatesR set of real numbersRn set of n-dimensional real vectors

∅ empty setd/d first order derivative with respect to d2/d2 second order derivative with respect to ∂/∂ first order partial derivative with respect to ∂2/∂2 second order partial derivative with respect to first order time derivative of a variable second order time derivative of a variable ∇ del operator∇2 Laplace operator∇ gradient of a vector field ∇ · divergence of a vector field ∇× curl of a vector field Re() real part of a variable Im() imaginary part of a variable ∗ complex conjugate of a variable

xxix

xxx List of Symbols

I identity matrix−1 inverse of a matrix T transpose of a vector or matrix H Hermitian or conjugate transpose of a vector or matrix det determinant of a matrix ei unit vector along axis if frequencyF [(); •] Fourier transformation of a function () from to •Hn[(); •] n-th order Hankel transformation of a function () from to •i imaginary unit

√−1

Jn() n-th order Bessel function of the first kindk wavenumberk dimensionless wavenumbern circumferential wavenumbert timeδij Kronecker Deltaδ() Dirac Delta functionω circular frequency

List of abbreviations

ARMA AutoRegressive Moving AverageCDF Cumulative Distribution FunctionCPT Cone Penetration TestEDT ElastoDynamics ToolboxFFT Fast Fourier TransformationPDF Probability Density FunctionP-wave Primary wave (dilatational wave)SASW Spectral Analysis of Surface WavesS-wave Secondary wave (shear wave)SH-wave Horizontally polarized S-waveSV-wave Vertically polarized S-waveSCPT Seismic Cone Penetration TestSEA Statistical Energy AnalysisSFEM Stochastic Finite Element Method

List of Symbols xxxi

The direct stiffness method for layered soils

aI incident wave potential amplitudes in the two-dimensional caseaR reflected wave potential amplitudes in the two-dimensional caseaI′ incident wave potential amplitudes in the three-dimensional caseaR′ reflected wave potential amplitudes in the three-dimensional caseA matrix relating three-dimensional to two-dimensional wave potentialsBI matrix relating displacements to incident wave potentialsBR matrix relating displacements to reflected wave potentialsBe traction shape functions for element eBe

PSV traction shape functions for in-plane waves in element eBe

SH traction shape functions for out-of-plane waves in element eC elasticity tensorCI matrix relating tractions to incident wave potentialsCR matrix relating tractions to reflected wave potentialsCn matrix describing the variation of the wave field with the distance rC′n matrix describing the variation of the strains with the distance r

Cp P-wave velocityCs S-wave velocityDI matrix relating modified displacements to incident wave potentialsDR matrix relating modified displacements to reflected wave potentialsE Young’s modulusGI matrix relating modified tractions to incident wave potentialsGR matrix relating modified tractions to reflected wave potentialsIP incident P-wave potential amplitude in the two-dimensional caseISH incident SH-wave potential amplitude in the two-dimensional caseISV incident SV-wave potential amplitude in the two-dimensional caseI ′P incident P-wave potential amplitude in the three-dimensional caseI ′SH incident SH-wave potential amplitude in the three-dimensional caseI ′SV incident SV-wave potential amplitude in the three-dimensional caseJIn matrix relating displacements to incident wave potentials

JRn matrix relating displacements to reflected wave potentials

kp P-wave propagation vectorks S-wave propagation vectorkp magnitude of the P-wave propagation vector kp

ks magnitude of the S-wave propagation vector ks

kx horizontal wavenumber in Cartesian coordinateskr horizontal wavenumber in cylindrical coordinateskzp vertical component of the P-wave propagation vector kp

kzs vertical component of the S-wave propagation vector ks

K stiffness matrix for a layered soilKPSV stiffness matrix for in-plane waves in a layered soilKSH stiffness matrix for out-of-plane waves in a layered soil

xxxii List of Symbols

Ke stiffness matrix for element eKe

PSV stiffness matrix for in-plane waves in element eKe

SH stiffness matrix for out-of-plane waves in element el factor to make the wave potentials of the same dimensionL layer element thicknessn unit outward normal vectorNe displacement shape functions for element eNe

PSV displacement shape functions for in-plane waves in element eNe

SH displacement shape functions for out-of-plane waves in element ep load vectorp modified load vectorRP reflected P-wave potential amplitude in the two-dimensional caseRSH reflected SH-wave potential amplitude in the two-dimensional caseRSV reflected SV-wave potential amplitude in the two-dimensional caseR′

P reflected P-wave potential amplitude in the three-dimensional caseR′

SH reflected SH-wave potential amplitude in the three-dimensional caseR′

SV reflected SV-wave potential amplitude in the three-dimensional cases ratio of the wave velocities Cs and Cp

tn tractions on a plane with unit outward normal vector n

tez traction vector on a horizontal planetez modified traction vector on a horizontal planetez

PSV modified in-plane tractions on a horizontal planetez

SH modified out-of-plane tractions on a horizontal planete modified traction vector on the boundaries of element etePSV modified in-plane tractions on the boundaries of element eteSH modified out-of-plane tractions on the boundaries of element eT matrix defining the modified displacements and tractionsTn matrix describing the variation of the wave field with the angle θT′n matrix describing the variation of the strains with the angle θ

u displacement vectoru modified displacement vectoruPSV modified in-plane displacementsuSH modified out-of-plane displacementsue modified displacement vector on the boundaries of element euePSV modified in-plane displacements on the boundaries of element eueSH modified out-of-plane displacements on the boundaries of element eZI matrix describing the variation of the incident wave field with depthZR matrix describing the variation of the reflected wave field with depthβp hysteretic material damping ratio for P-wavesβs hysteretic material damping ratio for S-wavesε small strain tensorλ first Lame constantµ second Lame constant (shear modulus)ν Poisson’s ratio

List of Symbols xxxiii

ρ densityρb body forcesσ Cauchy stress tensorΦ P-wave potentialΨ S-wave potentialχ SH-wave potentialΨ′ SV-wave potentialΨ SV-wave potential for two-dimensional wave propagation

Green’s functions of layered soils

Cmin lowest (shear) wave velocity in the soil modelCmax highest (dilatational) wave velocity in the soil modelk0 reference wavenumberkmin lower bound of the wavenumber range of interestkmax upper bound of the wavenumber range of interestr0 reference distancermin lower bound of the spatial interval of interestrmax upper bound of the spatial interval of interestR source-receiver distances dual variable of the variables v and w

tGez

ij Green’s traction tensor

tGez

ij modified Green’s traction tensor

uGij Green’s displacement tensor

uGij modified Green’s displacement tensorv auxiliary variable defined as ln(x/x0) or ln(r/r0)w auxiliary variable defined as − ln(kx/k0) or − ln(kr/k0)

wψmin lower bound of the window where the kernel ψ(w) is not smallwψmax upper bound of the window where the kernel ψ(w) is not smallx0 reference distancexmin lower bound of the spatial interval of interestxmax upper bound of the spatial interval of interestβmin lowest hysteretic material damping ratio in the soil modelΓ() Gamma function∆w sampling interval in logarithmic spaceε threshold value to consider the kernel ψ(w) as smallεGijk Green’s strain tensor

σGijk Green’s stress tensor

φ angle of the source-receiver line with respect to the z-axisϕ input function of the convolution in Talman’s algorithmχ output function of the convolution in Talman’s algorithmψ convolution kernel in Talman’s algorithm

xxxiv List of Symbols

Dynamic soil-structure interaction

fb modal force vectorH foundation-soil transfer functionKb modal structural stiffness matrixKs modal soil impedance matrixMb modal structural mass matrixnb unit outward normal vector on the boundary of the structural domainns unit outward normal vector on the boundary of the soil domaintnb

b tractions on the boundary of the structural domaintnss tractions on the boundary of the soil domain

ub displacement field in the structural domainus displacement field in the soil domainQ dimension of the kinematic basis used in the interaction problemΓ coherence of the hammer force and the free field responseΓbσ free boundary of the structural domainαm m-th modal coordinateΓsσ free boundary of the soil domainΓs∞ outer boundary of the soil domainσb stress field in the structural domainσs stress field in the soil domainΣ soil-structure interfaceψbm m-th basis vector for the displacements in the structural subdomainψsm m-th basis vector for the displacements in the soil subdomainΩb structural domainΩs soil domain

The SASW method

CR phase velocity of the Rayleigh wavesCL phase velocity of the Love wavesCE

R experimental dispersion curveCT

R theoretical dispersion curved thickness of a soil layerfobj objective functionHij transfer function between receivers i and jrmin minimum number of wavelengths between two receiversrmax maximum number of wavelengths between two receiversSkij cross power spectral density between receivers i and j for event kSij average cross power spectral density between receivers i and jΓij coherence between receivers

List of Symbols xxxv

Γmin coherence threshold∆rij distance between receiversθij unfolded phase of the transfer function Hij

Stochastic mechanics

(Ω, S, P ) probability space(θ) random vector(x, θ) random processBRn Borel σ-algebra over R

n

BH Borel σ-algebra over H

cijk expected value of the product of Ψi, Ψj , and Ψk

C covariance matrix of a random vector C covariance function of a random process CE exponential covariance functionCG Gaussian covariance functionCM Matern covariance functiondk mean square value of Ψk

D domain of a random process (index set)E mathematical expectationfk k-th Karhunen-Loeve modeg memoryless transformationF CDF of the random vector or process FG Gaussian CDFH vector space of functions f : D → R

n

K stiffness matrixlc correlation lengthm mean value of a random variable or process mp p-th order statistical moment of a random vector M order of Karhunen-Loeve decompositionP force vectorP probability measureP probability function of the random vector or process p PDF of the random vector or process pB beta PDFpG Gaussian PDFpL lognormal PDFpU uniform PDFr correlation coefficient matrix of a random vector R correlation matrix of a random vector R correlation function of a random process S event space

xxxvi List of Symbols

S eventU response vectorwGn Gaussian weights for Gauss-Hermite quadrature

ε2M mean square error for a Karhunen-Loeve decomposition of order Mηk underlying Gaussian variables used in the Nataf transformationη underlying Gaussian process used to model a translation processθ elementary event (coordinate in the random dimension)λk k-th eigenvalue of a correlation matrix or functionξk independent standard Gaussian variablesξGn Gaussian points for Gauss-Hermite quadratureσ standard deviation of a random variable or process Φk k-th eigenvector of a correlation matrixΦp p-th order one-dimensional Hermite polynomialΨi i-th multidimensional Hermite polynomialΩ sample space

Wave propagation in a soil with uncertain proper-

ties

Cµ covariance function of the random shear modulusL length of the domain of the random shear modulusmµ mean value of the shear modulusmν mean value of the Poisson’s ratiomβ mean value of the hysteretic material damping ratiomρ mean value of the densitymH mean value of the foundation-soil transfer functionm|H| mean value of the modulus of the foundation-soil transfer functionpµ PDF of the shear moduluspν PDF of the Poisson’s ratiopβ PDF of the hysteretic material damping ratiopρ PDF of the densityqc cone tip resistanceσµ standard deviation of the shear modulusσν standard deviation of the Poisson’s ratioσβ standard deviation of the hysteretic material damping ratioσρ standard deviation of the densityσH standard deviation of the foundation-soil transfer functionσ|H| standard deviation of the modulus of the transfer function

List of Symbols xxxvii

Stochastic soil characterization by means of the

SASW test

cξkcorrelation coefficient of the variable ξk in the Markov chain

Lξ likelihood functionmξk

mean value of the variable ξk in the Markov chainkz vertical wavenumberq proposal densityr acceptance probabilityr1 acceptance probability accounting for the prior PDFr2 acceptance probability accounting for the likelihoodW wavelet power spectrumξi i-th state in the Markov chainξ′i candidate for the i-th state in the Markov chainρξ prior PDFσξ posterior PDFσq standard deviation of the proposal densityσξk

standard deviation of the variable ξk in the Markov chainψ Paul wavelet

xxxviii List of Symbols

Contents

Voorwoord iii

Samenvatting v

Summary xxvii

List of Symbols xxix

Contents xxxix

List of Figures xliii

List of Tables li

1 Introduction 1

1.1 Vibrations in the built environment . . . . . . . . . . . . . . . . . . 11.1.1 Sources of vibrations . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Consequences of vibrations . . . . . . . . . . . . . . . . . . 2

1.2 Vibration predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Prediction models . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Identification of the dynamic soil properties . . . . . . . . . 71.2.3 Validation of the predictions . . . . . . . . . . . . . . . . . 9

1.3 Focus of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Solution strategy . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3 Original contributions . . . . . . . . . . . . . . . . . . . . . 16

1.4 A basic dynamic soil-structure interaction problem . . . . . . . . . 171.5 Organization of the text . . . . . . . . . . . . . . . . . . . . . . . . 19

2 The direct stiffness method for layered soils 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . 25

xxxix

xl Contents

2.2.2 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . 302.3 Two-dimensional wave propagation . . . . . . . . . . . . . . . . . . 32

2.3.1 Wave potentials . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.2 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.3 Tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3.4 Strains and stresses . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Three-dimensional wave propagation . . . . . . . . . . . . . . . . . 402.4.1 Wave potentials . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.2 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.3 Tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.4 Strains and stresses . . . . . . . . . . . . . . . . . . . . . . 47

2.5 Stiffness matrices and shape functions . . . . . . . . . . . . . . . . 482.5.1 Stiffness matrices . . . . . . . . . . . . . . . . . . . . . . . . 492.5.2 Shape functions for the displacements . . . . . . . . . . . . 552.5.3 Shape functions for the tractions . . . . . . . . . . . . . . . 56

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Green’s functions of layered soils 59

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Two-dimensional Green’s functions . . . . . . . . . . . . . . . . . . 61

3.2.1 Excitation in the x-direction . . . . . . . . . . . . . . . . . 613.2.2 Excitation in the y-direction . . . . . . . . . . . . . . . . . 633.2.3 Excitation in the z-direction . . . . . . . . . . . . . . . . . . 64

3.3 Three-dimensional Green’s functions . . . . . . . . . . . . . . . . . 663.3.1 Excitation in the x-direction . . . . . . . . . . . . . . . . . 663.3.2 Excitation in the y-direction . . . . . . . . . . . . . . . . . 683.3.3 Excitation in the z-direction . . . . . . . . . . . . . . . . . . 70

3.4 Numerical Fourier transformations . . . . . . . . . . . . . . . . . . 723.5 Numerical Hankel transformations . . . . . . . . . . . . . . . . . . 763.6 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Dynamic soil-structure interaction 87

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Subdomain formulation . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 The structural subdomain . . . . . . . . . . . . . . . . . . . 884.2.2 The soil subdomain . . . . . . . . . . . . . . . . . . . . . . 894.2.3 The interaction problem . . . . . . . . . . . . . . . . . . . . 89

4.3 Boundary integral formulation . . . . . . . . . . . . . . . . . . . . 914.3.1 The dynamic reciprocity theorem . . . . . . . . . . . . . . . 914.3.2 The boundary integral equations . . . . . . . . . . . . . . . 92

4.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.4.1 The structural modes . . . . . . . . . . . . . . . . . . . . . 934.4.2 The modal soil tractions . . . . . . . . . . . . . . . . . . . . 94

Contents xli

4.4.3 The foundation-soil impedance . . . . . . . . . . . . . . . . 954.4.4 The radiated wave field . . . . . . . . . . . . . . . . . . . . 964.4.5 The foundation-soil transfer functions . . . . . . . . . . . . 97

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 The SASW method 101

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Surface waves in a layered halfspace . . . . . . . . . . . . . . . . . 102

5.2.1 Rayleigh waves . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.2 Love waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 The experimental dispersion curve . . . . . . . . . . . . . . . . . . 1045.4 The theoretical dispersion curve . . . . . . . . . . . . . . . . . . . . 1085.5 The inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.6 The foundation-soil transfer functions . . . . . . . . . . . . . . . . 1105.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Stochastic mechanics 115

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Kolmogorov’s probability theory . . . . . . . . . . . . . . . . . . . 117

6.2.1 The probability space . . . . . . . . . . . . . . . . . . . . . 1176.2.2 Random variables . . . . . . . . . . . . . . . . . . . . . . . 1186.2.3 Random processes . . . . . . . . . . . . . . . . . . . . . . . 1216.2.4 Probability distributions . . . . . . . . . . . . . . . . . . . . 1246.2.5 Covariance functions . . . . . . . . . . . . . . . . . . . . . . 126

6.3 Random vector modelling . . . . . . . . . . . . . . . . . . . . . . . 1286.3.1 The decorrelation of random variables . . . . . . . . . . . . 1286.3.2 The Nataf transformation . . . . . . . . . . . . . . . . . . . 130

6.4 Random process modelling . . . . . . . . . . . . . . . . . . . . . . . 1316.4.1 The Karhunen-Loeve decomposition . . . . . . . . . . . . . 1336.4.2 Non-Gaussian translation processes . . . . . . . . . . . . . . 135

6.5 The stochastic finite element method . . . . . . . . . . . . . . . . . 1366.5.1 The perturbation method . . . . . . . . . . . . . . . . . . . 1386.5.2 The spectral stochastic finite element method . . . . . . . . 1406.5.3 The Monte Carlo method . . . . . . . . . . . . . . . . . . . 145

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7 Wave propagation in a soil with uncertain properties 149

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.2 Wave propagation in a homogeneous soil with uncertain properties 151

7.2.1 Uncertain shear modulus . . . . . . . . . . . . . . . . . . . 1517.2.2 Uncertain Poisson’s ratio . . . . . . . . . . . . . . . . . . . 1587.2.3 Uncertain damping ratio . . . . . . . . . . . . . . . . . . . . 1607.2.4 Uncertain density . . . . . . . . . . . . . . . . . . . . . . . . 161

xlii Contents

7.3 Wave propagation in an inhomogeneous soil with an uncertain shearmodulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.3.1 The stochastic soil model . . . . . . . . . . . . . . . . . . . 1637.3.2 The foundation-soil transfer functions . . . . . . . . . . . . 1667.3.3 Influence of the domain of the random process . . . . . . . 1687.3.4 Influence of the order of the Karhunen-Loeve decomposition 1697.3.5 Influence of the correlation length . . . . . . . . . . . . . . 1717.3.6 Influence of the marginal PDF . . . . . . . . . . . . . . . . 173

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8 Stochastic soil characterization by means of the SASW test 177

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.2 The prior model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.3 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . 1828.4 The Bayesian updating scheme . . . . . . . . . . . . . . . . . . . . 1848.5 The posterior model . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.6 The foundation-soil transfer functions . . . . . . . . . . . . . . . . 1908.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

9 Conclusions and recommendations for further research 195

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Recommendations for further research . . . . . . . . . . . . . . . . 197

Bibliography 201

Curriculum vitae 221

List of Figures

1 De meetopstelling op de site in Lincent. . . . . . . . . . . . . . . . viii2 Modulus van de fundering-grondtransferfunctie H(ω) op (a) 4 m,

(b) 8 m, (c) 16 m, en (d) 32 m van het midden van de fundering. Eenvolle lijn geeft aan dat de coherentie Γ(ω) de waarde Γmin = 0.95overschrijdt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

3 Experimentele dispersiecurve (grijze punten) en polynomiale be-nadering (zwarte punten) voor de site in Lincent. . . . . . . . . . . x

4 Experimentele (zwarte punten) en theoretische (grijze lijn) disper-siecurve voor de site in Lincent. . . . . . . . . . . . . . . . . . . . . xi

5 Modulus van de gemeten (zwarte lijn) en voorspelde (grijze lijn)fundering-grondtransferfunctie H(ω) op (a) 4 m, (b) 8 m, (c) 16 m,en (d) 32 m van het midden van de fundering. Een volle zwarte lijngeeft aan dat de coherentie Γ(ω) de waarde Γmin = 0.95 overschrijdt. xiii

6 A priori (a) marginale PDF pµ(µ) en (b) covariantiefunctie Cµ(z)van de glijdingsmodulus µ(z, θ). . . . . . . . . . . . . . . . . . . . . xv

7 (a) Tien realisaties van de a priori glijdingsmodulus µ(z, θ) en (b)overeenkomstige theoretische dispersiecurves CT

R(ω) (grijze lijnen),vergeleken met de experimentele dispersiecurve CE

R(ω) (zwartepunten). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

8 (a) Tien realisaties van de a posteriori glijdingsmodulus µ(z, θ)en (b) overeenkomstige theoretische dispersiecurves CT

R (ω) (grijzelijnen), vergeleken met de experimentele dispersiecurve CE

R(ω)(zwarte punten). . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

9 Tien realisaties (grijze lijnen) en 95 % betrouwbaarheidsinterval(grijze zone) van de modulus van de a priori transferfunctie H(ω)vergeleken met de experimentele data (zwarte lijn) op (a) 4 m,(b) 8 m, (c) 16 m, en (d) 32 m van het midden van de fundering.Een volle zwarte lijn geeft aan dat de coherentie Γ(ω) de waardeΓmin = 0.95 overschrijdt. . . . . . . . . . . . . . . . . . . . . . . . . xviii

xliii

xliv List of Figures

10 Tien realisaties (grijze lijnen) en 95 % betrouwbaarheidsinterval(grijze zone) van de modulus van de a posteriori transferfunctieH(ω) vergeleken met de experimentele data (zwarte lijn) op (a) 4 m,(b) 8 m, (c) 16 m, en (d) 32 m van het midden van de fundering.Een volle zwarte lijn geeft aan dat de coherentie Γ(ω) de waardeΓmin = 0.95 overschrijdt. . . . . . . . . . . . . . . . . . . . . . . . . xix

11 Organisatie van de tekst. . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1.1 Measured (black line) and predicted (gray line) vertical free fieldvelocity at (a) 8 m and (b) 24 m from the center of the road due tothe passage of a truck on an artificial unevenness. . . . . . . . . . . 10

1.2 Measured (black line) and predicted (gray line) vertical free fieldvelocity at (a) 8 m and (b) 24 m from the track due to the passageof a high speed train. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Measurement site at the Rue de la Bruyere in Lincent. . . . . . . . 171.4 Location of the measurement line. . . . . . . . . . . . . . . . . . . 181.5 (a) The concrete foundation and the impact hammer and (b) an

accelerometer in the free field. . . . . . . . . . . . . . . . . . . . . . 191.6 Modulus of the foundation-soil transfer function H(ω) (left hand

side) and coherence function Γ(ω) (right hand side) for differentreceivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m fromthe center of the foundation. The modulus of the transfer functionH(ω) is plotted as a solid line if the coherence function Γ(ω) exceedsa threshold value Γmin = 0.95. . . . . . . . . . . . . . . . . . . . . . 20

1.7 Organization of the text. . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 (a) Dilatational and (b) shear wave. . . . . . . . . . . . . . . . . . 282.2 (a) Vertically propagating and (b) vertically evanescent P-wave. . . 352.3 The halfspace element. . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 The layer element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.5 A horizontally layered halfspace consisting of N − 1 layers on a

halfspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.6 Equilibrium of interface i between elements i− 1 and i. . . . . . . 54

3.1 (a) Window applied to the function ϕ(w) and (b) filter applied tothe kernel Ψ(s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Kernel ψ(w) of the logarithmic Fourier cosine transformation. . . . 753.3 Kernel ψ(w) of the logarithmic Hankel transformation. . . . . . . . 793.4 Real (black line) and imaginary (gray line) part of the wavenumber

domain Green’s function uGzz(z

′ = 0, kr, θ = 0, z = 0, ω) at 25 Hz. . 803.5 Real (black line) and imaginary (gray line) part of the (a) regular

and (b) singular component of the wavenumber domain Green’sfunction uG

zz(z′ = 0, kr, θ = 0, z = 0, ω) at 25 Hz. . . . . . . . . . . . 81

3.6 Real (black line) and imaginary (gray line) part of the function ϕ(w). 82

List of Figures xlv

3.7 Real (black line) and imaginary (gray line) part of the function χ(w). 823.8 Real (black line) and imaginary (gray line) part of the (a) regular

and (b) singular component of the space domain Green’s functionuGzz(z

′ = 0, r, θ = 0, z = 0, ω) at 25 Hz. . . . . . . . . . . . . . . . . 823.9 Real (black line) and imaginary (gray line) part of the regular

component of the space domain Green’s function uGzz(z

′ = 0, r, θ =0, z = 0, ω) at 25 Hz, calculated without window or filter. . . . . . . 83

3.10 Real (black line) and imaginary (gray line) part of the space domainGreen’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω) at 25 Hz . . . . . . . 83

3.11 Real part of the Green’s functions uGzj(z

′ = 0, r, θ = 0, z, ω) at 25 Hz. 84

3.12 Modulus of the Green’s function uGzz(z

′ = 0, r, θ = 0, z = 0, ω) for asource-receiver distance r of (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m. 85

4.1 Geometry of the subdomains. . . . . . . . . . . . . . . . . . . . . . 884.2 First nine eigenmodes of the concrete foundation. Modes (a) to (f)

are rigid body modes. The eigenfrequencies corresponding to theother modes are (g) 1452 Hz, (h) 2084 Hz, and (i) 2357 Hz. . . . . . 94

4.3 Real (top) and imaginary (bottom) part of the modal soil tractions(a) tns

sx(ψs1), (b) tnssy (ψs1), and (c) tns

sz (ψs1) at 100 Hz. . . . . . . . 954.4 Real (black line) and imaginary (gray line) part of (a) the structural

impedance[

Kb − ω2Mb

]

11and (b) the soil impedance [Ks]11. . . 95

4.5 Modulus of the foundation-soil impedance[

Kb − ω2Mb + Ks

]

11. . 96

4.6 (a) Real and (b) imaginary part of the radiated wave field ψs1 at100 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.7 Modulus of the foundation-soil transfer function H(ω) (black line)and the Green’s function uG

zz (gray line) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the centerof the foundation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.8 Modulus of the foundation-soil transfer function H(ω) for differentreceivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m fromthe center of the foundation. The gray line is calculated with arefined model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1 Phase velocity of the (a) Rayleigh and (b) Love waves in a layeredhalfspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 (a) First, (b) second, and (c) third Rayleigh wave and (d) first, (e)second, and (f) third Love wave in a layered halfspace at 80 Hz. . . 105

5.3 Time history of the free field acceleration due to a hammer impacton the concrete foundation as a function of the distance from thesource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Experimental dispersion curve (gray dots) and approximatingpolynomial (black dots) obtained for the site in Lincent. . . . . . . 107

xlvi List of Figures

5.5 Effective theoretical dispersion curve CTR (ω) obtained from the

wavenumber domain Green’s function uGzz(z

′ = 0, kr, z = 0, ω) (grayline) compared with a reference solution derived from the numericalsimulation of an SASW test (black dots) and with the theoreticaldispersion curves of all modes (thin black lines) for (a) soil profile1 and (b) soil profile 2. . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.6 Experimental (black dots) and theoretical (gray line) dispersioncurve for (a) a homogeneous halfspace, (b) a layer on a halfspace,(c) two layers on a halfspace, and (d) three layers on a halfspace. . 111

5.7 Modulus of the measured (black line) and predicted (gray line)foundation-soil transfer function H(ω) for different receivers, lo-cated at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the centerof the foundation. The modulus of the measured transfer functionH(ω) is plotted as a solid line if the coherence function Γ(ω) betweenthe hammer force and the free field response exceeds a thresholdvalue Γmin = 0.95. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1 Kolmogorov’s representation of a random vector v : Ω → Rn. . . . 118

6.2 Real (black line) and imaginary (gray line) part of the foundation-soil transfer function H(ω) at 32 m from the center of the foundationand at (a) 0 Hz and (b) 100 Hz. . . . . . . . . . . . . . . . . . . . . 137

7.1 Probability density function pµ(µ) of the random shear modulus µ(θ).1517.2 Ten realizations of the real part of the foundation-soil transfer

function H(ω) for different receivers, located at (a) 4 m, (b) 8 m,(c) 16 m, and (d) 32 m from the center of the foundation. Darkerlines correspond to higher values of the random shear modulus µ(θ). 152

7.3 Ten realizations of the imaginary part of the foundation-soil transferfunction H(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c)16 m, and (d) 32 m from the center of the foundation. Darker linescorrespond to higher values of the random shear modulus µ(θ). . . 153

7.4 Ten realizations of the modulus of the foundation-soil transferfunction H(ω) for different receivers, located at (a) 4 m, (b) 8 m,(c) 16 m, and (d) 32 m from the center of the foundation. Darkerlines correspond to higher values of the random shear modulus µ(θ). 154

7.5 Modulus of the mean value mH(ω) (thick line) and standarddeviation σH(ω) (thin line) of the foundation-soil transfer function

H(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m,and (d) 32 m from the center of the foundation. The soil is modelledas a homogeneous halfspace with a random shear modulus µ(θ). . . 155

List of Figures xlvii

7.6 Mean value (thick line) and standard deviation (thin line) of themodulus of the foundation-soil transfer function H(ω) for differentreceivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m fromthe center of the foundation. The soil is modelled as a homogeneoushalfspace with a random shear modulus µ(θ). . . . . . . . . . . . . 156

7.7 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. Darker lines correspond tohigher values of the random shear modulus µ(θ). . . . . . . . . . . 157

7.8 Five estimations of the 95 % confidence region of the modulus ofthe foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the centerof the foundation. The soil is modelled as a homogeneous halfspacewith a random shear modulus µ(θ). . . . . . . . . . . . . . . . . . . 158

7.9 Probability density function pν(ν) of the random Poisson’s ratio ν(θ).1597.10 Ten realizations (gray lines) and 95 % confidence region (gray area)

of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. Darker lines correspond tohigher values of the random Poisson’s ratio ν(θ). . . . . . . . . . . 159

7.11 Probability density function pβ(β) of the random material dampingratio β(θ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.12 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. Darker lines correspond tohigher values of the random material damping ratio β(θ). . . . . . 161

7.13 Probability density function pρ(ρ) of the random density ρ(θ). . . . 1627.14 Ten realizations (gray lines) and 95 % confidence region (gray area)

of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. Darker lines correspond tohigher values of the random density ρ(θ). . . . . . . . . . . . . . . 162

7.15 (a) Marginal PDF pµ(µ) and (b) covariance function Cµ(z) of therandom shear modulus µ(z, θ). . . . . . . . . . . . . . . . . . . . . 164

7.16 Cone tip resistance qc(z) recorded in Sint-Katelijne-Waver. . . . . 1647.17 (a) Eigenvalues λk and (b) eigenfunctions fk(z) corresponding to

the lowest order modes in the Karhunen-Loeve decomposition ofthe underlying Gaussian process η(z, θ). In figure (b), darker linesrepresent modes of lower order. . . . . . . . . . . . . . . . . . . . . 165

7.18 Realization of the random shear modulus µ(z, θ) with a correlationlength lc = 0.25 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

xlviii List of Figures

7.19 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. The soil is modelled as aninhomogeneous halfspace with a random shear modulus µ(z, θ). . . 167

7.20 Realization of the modulus of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m,and (d) 32 m from the center of the foundation. The soil is modelledas an inhomogeneous halfspace with a random shear modulus µ(z, θ)that varies up to a depth L′ = 4 m (light gray lines), L′ = 8 m (darkgray lines), and L′ = 16 m (black lines). . . . . . . . . . . . . . . . 168

7.21 Realization of the random shear modulus µ(z, θ) obtained with aKarhunen-Loeve decomposition of order M ′ = 8 (light gray line),M ′ = 16 (dark gray line), and M ′ = 256 (black line). . . . . . . . . 169

7.22 Realization of the modulus of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m,and (d) 32 m from the center of the foundation. The soil is modelledas an inhomogeneous halfspace with a random shear modulusµ(z, θ), using a Karhunen-Loeve decomposition of order M ′ = 8(light gray lines), M ′ = 32 (dark gray lines), and M ′ = 256 (blacklines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.23 Covariance function Cµ(z) of the random shear modulus µ(z, θ)with a correlation length (a) lc = 0.125 m and (b) lc = 0.5 m. . . . 171

7.24 Realization of the random shear modulus µ(z, θ) with a correlationlength (a) lc = 0.125 m and (b) lc = 0.5 m. . . . . . . . . . . . . . . 171

7.25 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. The soil is modelled as aninhomogeneous halfspace with a random shear modulus µ(z, θ) witha correlation length lc = 0.125 m. . . . . . . . . . . . . . . . . . . . 172

7.26 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. The soil is modelled as aninhomogeneous halfspace with a random shear modulus µ(z, θ) witha correlation length lc = 0.5 m. . . . . . . . . . . . . . . . . . . . . 173

7.27 Uniform marginal PDF pµ(µ) of the random shear modulus µ(z, θ). 1747.28 Realization of the random shear modulus µ(z, θ), using a uniform

probability distribution pµ(µ). . . . . . . . . . . . . . . . . . . . . . 174

List of Figures xlix

7.29 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the foundation-soil transfer function H(ω) fordifferent receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. The soil is modelled as aninhomogeneous halfspace with a random shear modulus µ(z, θ) witha uniform probability distribution pµ(µ). . . . . . . . . . . . . . . . 175

8.1 (a) Prior marginal PDF pµ(µ) and (b) prior covariance functionCµ(z) of the soil’s dynamic shear modulus µ(z, θ). . . . . . . . . . 181

8.2 (a) Ten realizations of the shear modulus µ(z, θ) drawn from theprior stochastic soil model and (b) corresponding theoretical dis-persion curves CT

R (ω) (gray lines), compared with the experimentaldispersion curve CE

R(ω) (black dots). . . . . . . . . . . . . . . . . . 1828.3 (a) Two realizations of the random shear modulus µ(z, θ) and

(b) corresponding theoretical dispersion curves CTR(ω), compared

with the experimental dispersion curve CER(ω) (black dots) and the

bounds imposed by the likelihood function Lξ(ξ) (gray area). Theblack line corresponds to an acceptable soil profile, the gray line toan unacceptable soil profile. . . . . . . . . . . . . . . . . . . . . . . 183

8.4 (a) Shear modulus and (b) dispersion curve for ten successivecandidate states in the Markov chain. Black lines correspondto acceptable candidates, gray lines correspond to unacceptablecandidates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.5 (a) Mean and (b) standard deviation of the random variables ξk(θ)in the Markov chain. Darker lines correspond to lower orderKarhunen-Loeve modes. . . . . . . . . . . . . . . . . . . . . . . . . 187

8.6 Correlation coefficient cξk(∆i) of the random variables ξk(θ) in the

Markov chain. Darker lines correspond to lower order Karhunen-Loeve modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.7 (a) Ten realizations of the shear modulus µ(z, θ) drawn from theposterior stochastic soil model and (b) corresponding theoreticaldispersion curves CT

R (ω) (gray lines), compared with the experi-mental dispersion curve CE

R(ω) (black dots). . . . . . . . . . . . . . 1888.8 Real (black line) and imaginary (gray line) part of a fourth order

Paul wavelet ψ(z). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.9 (a) Prior and (b) posterior wavelet power spectrum |W (z, kz)|2 of

the soil’s dynamic shear modulus µ(z, θ). The hatched area is thecone of influence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.10 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the prior transfer function H(ω) compared withthe experimental data (black line) for a source-receiver distanceof (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m. The modulus of themeasured transfer function H(ω) is plotted as a solid line if thecoherence function Γ(ω) exceeds a threshold value Γmin = 0.95. . . 191

l List of Figures

8.11 Ten realizations (gray lines) and 95 % confidence region (gray area)of the modulus of the posterior transfer function H(ω) comparedwith the experimental data (black line) for a source-receiver distanceof (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m. The modulus of themeasured transfer function H(ω) is plotted as a solid line if thecoherence function Γ(ω) exceeds a threshold value Γmin = 0.95. . . 192

List of Tables

1 Grondprofiel afgeleid uit de SASW-proef. . . . . . . . . . . . . . . xi

5.1 Soil profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Benchmark soil profile 1. . . . . . . . . . . . . . . . . . . . . . . . . 1085.3 Benchmark soil profile 2. . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Initial and final soil profile for all stages in the inversion procedure. 1115.5 Soil profile used for the numerical vibration predictions. . . . . . . 112

li

lii List of Tables

Chapter 1

Introduction

1.1 Vibrations in the built environment

Ground-borne vibrations in the built environment are a matter of growing envi-ronmental concern, especially in densely populated areas as Flanders. Increasingdynamic impacts of different nature combined with a growing environmentalconcern and the enhancement of a sustainable development policy, have resultedin an increased interest and awareness for the problem of vibrations in the builtenvironment among the population and local and federal authorities.

1.1.1 Sources of vibrations

Traffic induced vibrations are a major concern. An important trigger has been theintroduction of the high speed train network throughout Europe and the extensionof local rail (both passenger and freight trains) and subway infrastructure.Examples in Belgium are the HST tracks L1 and L2 that link Brussels withParis and Koln, and the HST track L4 that links Brussels with the Netherlandsand involves the north-south crossing of Antwerp in two bored tunnels. Otherexamples are the railway tunnel at Brussels airport and the suburban light railnetwork around Brussels.

As to road traffic, the dynamic impact has also considerably increased. Reduc-tion of stock and just-in-time delivery policies have contributed to a substantialincrease of freight transport by trucks. The regional road administration inFlanders is confronted with an increasing number of complaints, especially onmain roads with a pavement consisting of discrete concrete plates with joints,as well as town crossings where speed reducing infrastructure has been installed,resulting in increased vibration levels.

Besides traffic induced vibrations, vibrations due to construction and industrialactivities are also a matter of growing concern. There is an increasing demand, forexample, to use vibratory pile driving as an alternative to impact pile driving in

1

2 Introduction

densely populated areas. Vibrations caused by blast loading due to the controlledexplosion of mines, left behind after the first and second World War in shallowoffshore waters along the Belgian coast, are also a potential threat for nearbybuildings. Industrial machinery as looms and printing presses is also responsiblefor high vibration levels.

1.1.2 Consequences of vibrations

Foreign norms and guidelines recognize malfunctioning of sensitive equipment [81,208], discomfort to people [27, 50, 105, 207], and damage to buildings [51, 206] aspossible consequences of vibrations in buildings.

Maximum vibration levels with respect to sensitive equipment are usuallyspecified in terms of one-third octave band velocity spectra. Dependent on the typeof equipment, the maximum root mean square velocity varies from 0.003 mm/s to0.2 mm/s in the one-third octave bands from 8 Hz to 80 Hz [81, 208].

In the frequency range between 1 Hz and 80 Hz, people in buildings perceivevibrations as a mechanical vibration of the human body, while at higher frequenciesbetween 25 Hz and 200 Hz, vibrations can also be perceived as re-radiated noise,emitted by vibrating parts of the building. The threshold value for the directperception of vibrations is of the order of magnitude of 0.1 mm/s. Complaintsabout discomfort are often related to the fear for a reduction of serviceability anda loss of economic value of the building.

Damage to buildings is generally described as a change of the properties orposition of a structural member, resulting in one of the following consequences:(1) failure of a structural member, (2) decrease of the structural integrity ofa member or the whole structure, resulting in a significant reduction of theserviceability (decrease of lifetime), and (3) a loss of esthetic and economic valuedue to cracks in walls and ceilings, separation of masonry blocks, and cracks in tiles.Damage to buildings occurs at high vibration levels compared to the thresholdvalue for human perception. Dependent on the type of the building, the criticalpeak particle velocity varies from 3 mm/s to 20 mm/s at the foundation level andfrom 8 mm/s to 40 mm/s (in the horizontal direction) at the highest level of thebuilding [51, 206].

1.2 Vibration predictions

During the last century, numerous empirical and numerical prediction models forground-borne vibrations have been developed. Empirical models are based onpractical rules of thumb inferred from engineering experience or experimentaldata. Such rules of thumb are valuable tools in engineering practice, but theycan only be used in situations similar to those for which the measurements havebeen made. Theoretical models start from the laws of physics. Dependent on theassumptions made in the further development of these models, they are applicable

Vibration predictions 3

to a wide range of situations. Theoretical vibration prediction models are thereforewell suited to assess vibration reduction measures in existing and new situations.Moreover, these models allow to investigate the physical phenomena governing thevibration problem. In the following, a selection of theoretical vibration predictionmodels is briefly reviewed.

1.2.1 Prediction models

Ground-borne vibrations in the built environment are caused by dynamic forcesacting on structures (the source) such as roads, tracks, tunnels, piles, . . . Theseforces are transmitted to the soil where they induce elastodynamic waves. Thewaves impinge on the foundations of nearby buildings (the receiver), where theygenerate structural vibrations and re-radiated noise.

The prediction of ground-borne vibrations in buildings is a dynamic soil-structure interaction problem consisting of three subproblems: the characteriza-tion of the source, the transmission of waves through the soil, and the interactionof the receiver with the incident wave field. Usually, the dynamic soil-structureinteraction problem is solved in two steps. In the first step, a source model is usedto predict the free field vibrations radiated by the source. In the second step, thefree field vibrations are applied to a receiver model as an incident wave field. Thisapproach is accurate if the wavelength of the waves in the soil is small comparedto the source-receiver distance.

Road traffic induced vibrations

Road traffic induced vibrations are mainly caused by the moving axle loads ofheavy trucks passing on roads with an uneven surface. The axle loads consist of astatic component and a dynamic component. The static component is due to theweight of the vehicle. If the vehicle speed is low compared to the wave velocitiesin the soil, the static component of the axle loads does not generate waves andits contribution to the free field response is negligible [94, 100]. The dynamiccomponent is due to the interaction between the vehicle and the road.

Two groups of vehicle eigenmodes dominate the dynamic axle loads: the pitchand bounce modes at relatively low eigenfrequencies between 0.8 Hz and 3 Hz andthe axle hop modes at frequencies between 8 Hz and 15 Hz [132]. The axle hopmodes play a crucial role in the generation of traffic-induced vibrations [6, 7].

Due to the large stiffness of the road compared to the stiffness of the tyres andthe vehicle’s suspension system, the effect of the road displacements on the axleloads is negligible [30, 80, 133]. The vehicle-road interaction problem can thereforebe uncoupled from the road-soil interaction problem. In the first subproblem, theaxle loads are calculated from the road unevenness. In the second subproblem,the free field response is computed from the axle loads.

4 Introduction

Several authors apply the axle loads directly to the soil in order to computethe free field response, neglecting the dynamic interaction of the road and the soil[40, 96, 101, 154].

Lombaert [132] uses a more advanced source model based on a subdomainformulation for dynamic soil-structure interaction developed by Aubry andClouteau [16, 36]. The road is modelled as an infinite elastic beam with a rigid crosssection. The soil is modelled by means of a boundary element formulation basedon the Green’s functions of a layered soil. The Green’s functions are calculated bymeans of the direct stiffness method [113]. The interaction problem is solved inthe frequency-wavenumber domain, exploiting the longitudinal invariance of thecoupled road-soil system.

Train induced vibrations

Train induced vibrations are caused by several mechanisms, such as (1) quasi-staticexcitation due to moving axle loads, (2) wheel and rail roughness, (3) additionalwheel and rail defects, (4) parametric excitation due to the discrete support ofa track by sleepers, (5) discontinuities of the track such as switches and railjoints, (6) vehicle suspension, (7) spatial variation of the rail properties, (8) lateralloads due to vehicle guidance on tight radius of curvature, (9) acceleration anddeceleration of the vehicle, and (10) environmental conditions affecting wear andhence vibrations [106].

The deflections of the track have an important influence on the dynamic axleloads, so that the interaction between train, track, and soil has to be fullyaccounted for [102]. Furthermore, the train speed can be close to the wavevelocities in the soil. In such cases, the contribution of the static componentof the axle loads to the response in the free field can not be neglected [4].

Dieterman and Metrikine [52] and Metrikine and Popp [147] calculate theresponse of a track subjected to a moving load. The track is modelled as anelastic beam supported by an elastic halfspace. The track-soil interaction problemis simplified, neglecting the shear stresses at the track-soil interface and assuminga uniform distribution of normal stresses along the cross section of the track.Furthermore, the continuity of displacements is only enforced at the beam axis.The soil impedance is calculated analytically for the case of a homogeneoushalfspace [52] and a layer built in at its base [147]. Recently, Steenbergen andMetrikine [204] have investigated the effect of the interface conditions on thedynamic response to a moving load of a beam on a halfspace. They concludethat the simplified models give accurate results as long as the wavelength of thewaves in the soil is large compared to the beam width.

Hall [93] uses the finite element method to calculate the quasi-static responsedue to a sequence of moving axle loads. Both a two-dimensional and a three-dimensional finite element model are used.

Paolucci et al. [160] follow a similar approach, but use spectral elements (withLagrange polynomials as shape functions) to model the soil. The track is not

Vibration predictions 5

included in the model. Instead, the spatial distribution of the loading on the soilis analytically obtained from the response of a beam on elastic foundation. Two-dimensional and three-dimensional analyses are performed. From comparison withexperimental data, it is observed that the two-dimensional analysis leads to anoverestimation of the far field response, which is due to an underestimation ofradiation damping into the soil [90].

Kaynia et al. [115] and Madshus and Kaynia [139] apply a substructure methodto study the track-soil interaction problem. The track is modelled by coupledbeam elements. The impedance of the soil is calculated by means of the thin layermethod [112], using the response of the soil to a disc load. The coupling of thetrack to the soil is enforced at the nodes of the beam elements. The model onlyincorporates the quasi-static excitation.

Celebi [31] compares the substructure method applied by Kaynia et al. [115] andMadshus and Kaynia [139] to a substructure method where the soil is modelledwith a boundary element formulation. Both methods are used to assess the effectof an open trench next to the track.

Clouteau [37] and Aubry et al. [17] study the interaction of an infinite beamwith a horizontally layered elastic halfspace. The interaction problem is efficientlysolved in the frequency-wavenumber domain.

A similar model is developed by Lombaert et al. [136]. This model accounts forthe quasi-static contribution of the axle loads and for the dynamic excitation duewheel and rail roughness. It has been used to investigate the mitigation of ground-borne vibrations from railway traffic by means of continuous floating slabs [137].

Subway induced vibrations

For the prediction of subway induced vibrations, a dynamic vehicle-track-tunnel-soil interaction problem has to be solved. Two-dimensional finite element models[34, 35, 180] and coupled finite element-boundary element models [223] have beenused for this purpose. However, two-dimensional models are not suitable forpredicting road or railway traffic induced vibrations as they do not account for wavepropagation in the direction of the track and underestimate radiation dampinginto the soil [90]. The use of three-dimensional models for the prediction of trafficinduced vibrations is computationally expensive. Therefore, models have beendeveloped that exploit the invariance of the system in the longitudinal direction.

Forrest and Hunt [70], Hussein [103] and Hussein and Hunt [104] use a semi-analytical pipe-in-pipe model where the tunnel and the surrounding soil aremodelled as concentric thick cylindrical shells. A floating slab track is modelledas an Euler-Bernoulli beam and is coupled to the tunnel wall. Calculations withthis model are performed in the frequency-wavenumber domain.

Yang et al. [228] use a finite/infinite element method to model the coupledtunnel-soil system. The response due to a moving harmonic load in the tunnel iscalculated in the frequency-wavenumber domain.

6 Introduction

Stamos and Beskos [203] and Sheng et al. [194] solve the dynamic interactionproblem in the frequency-wavenumber domain by means of a coupled finiteelement-boundary element method.

Clouteau et al. [39], Degrande et al. [47], and Gupta et al. [90] assume thegeometry to be periodic in the longitudinal direction. They use a Floquettransformation of the longitudinal coordinate and solve the problem in thefrequency-wavenumber domain.

Vibrations from construction activities

Apart from traffic, a wide range of construction activities such as demolition works,impact and vibratory pile driving, vibratory soil compaction, and blast loadinginduces vibrations in the built environment. A good overview on vibrations dueto construction activities and blast loading is given by Dowding [56, 57].

Ma et al. [138] propose a method to predict the damage area, plastic zone andground motions generated by underground explosions. They use a dynamic failuremodel for the rock mass, considering uncoupled elastic damage and plastic flow ofrock material. The model is based on the piecewise linear Drucker-Prager strengthcriterion and accounts for the degradation of rock strength and Young’s moduluswith damage [121]. Wang et al. [224] present a non-linear model for the simulationof blast wave propagation in the soil. The soil is modelled as a three-phase materialbased on an elasto-plastic solid skeleton with damage dependent characteristics.

Other authors focus on the far field vibrations and disregard the non-linearconstitutive behaviour of the soil in the immediate vicinity of the source. Ramshawet al. [171] use a linear axisymmetric finite/infinite element model for the predictionof outgoing waves due to pile driving. Jongmans [108] suggests to compute thefree field response due to pile driving by the convolution of the Green’s function ofthe soil and an equivalent linear source function representing the vertical force atthe pile toe. The equivalent source function is determined from a vibration recordclose to the pile and the Green’s function. Masoumi et al. [143] use a coupledfinite element-boundary element method to solve the dynamic pile-soil interactionproblem.

Vibrations and re-radiated noise in buildings

The free field vibrations obtained from the source models can be used as theinput for a subsequent dynamic soil-structure interaction analysis to calculate theresponse of buildings. The problem is similar to those encountered in earthquakeengineering, where the response of a structure is calculated due to an incident wavefield. The frequency content of the vibrations considered (up to 200 Hz), however,is an order of magnitude higher than the frequency content of typical earthquakes(0 Hz to 10 Hz) [165].

Pyl et al. [165, 168, 169] use the subdomain formulation developed by Aubryand Clouteau [16, 36] to calculate the vibrations in a building due to the passage

Vibration predictions 7

of a truck on a nearby road. The building is modelled by means of the finiteelement method. The soil is modelled with the boundary element method, usingthe Green’s functions of a layered halfspace. Due to the relatively high frequencycontent of the excitation, fictitious eigenfrequencies arise in the boundary elementmodel [25, 54]. These fictitious eigenfrequencies are mitigated [166] by means of amethod developed in the field of acoustics [28]. A similar dynamic soil-structureinteraction model is used by Fiala et al. [68] to calculate the structural vibrationsin buildings due to rail traffic. In an additional computation step, they calculatethe re-radiated noise in the building from the structural vibrations by means ofan acoustic spectral finite element formulation [91].

1.2.2 Identification of the dynamic soil properties

The transmission of waves through the soil is an essential element of dynamic soil-structure interaction problems. Wave propagation in the soil is determined by thedynamic soil properties. Crucial parameters are the dynamic shear modulus andthe damping ratio of the soil. As a consequence, the knowledge of these dynamicsoil properties is essential for an accurate prediction of ground-borne vibrations.

The dynamic soil properties

The soil is most often modelled as an elastodynamic medium. The wave fieldin a homogeneous elastodynamic medium can be expressed as a superposition ofplane waves [5]. Two types of plane waves exist: dilatational and shear waves.In the dilatational (or: longitudinal, irrotational, primary, P) waves, the particlesmove parallel to the wave propagation direction. In the shear (or: transverse,equivoluminal, rotational, secondary, S) waves, the particles move perpendicularto the wave propagation direction. The phase velocities are Cp =

√

(λ + 2µ)/ρ

for the dilatational waves and Cs =√

µ/ρ for the shear waves, where λ and µ arethe Lame coefficients and ρ is the density. In a homogeneous medium, dilatationaland shear waves do not interact.

For the prediction of ground-borne vibrations, the soil is frequently modelledas a (layered) halfspace. In a (layered) halfspace, free surface waves exist [2, 5].The free surface waves are the natural modes of vibration of the halfspace. Theyresult from an interaction between P-waves and S-waves due to the presence ofthe free surface and the interfaces between layers. Surface waves propagate in thehorizontal direction and vanish with depth. For low frequencies, the wavelengthof the surface waves is large and the surface waves reach deep soil layers. Theselayers are generally stiff, resulting in a high phase velocity. For high frequencies,the wavelength of the surface waves is smaller and the surface waves travel throughshallow soil layers. These layers are generally softer, resulting in a lower phasevelocity. As a consequence, surface waves are dispersive: their phase velocityvaries with the frequency and is determined by the variation of the soil propertieswith depth.

8 Introduction

As waves propagate through the medium, their amplitude decreases. Thisattenuation is due to geometrical damping and material damping. Geometricalor radiation damping is due to the expansion of the wavefronts, resulting in thespreading of energy over an increasing area. In a homogeneous halfspace, thegeometrical attenuation of waves due to point sources is proportional to rn, with rthe travel distance and n = −0.5 for Rayleigh waves, n = −1 for body wavesat depth, and n = −2 for body waves along the surface [174]. Geometricalattenuation is not directly related to the material properties of the medium.Material damping is related to the dissipation of energy. Energy is dissipatedthrough several mechanisms, such as friction between solid particles in the skeletonand relative motion between the skeleton and the pore fluid. Their combined effectis represented by a material damping model. In structural mechanics, a viscousdamping model is frequently used. The effect of viscous damping can be explainedby means of a sprung mass system subjected to a harmonic displacement withunit amplitude. The energy dissipated by the system per cycle is proportionalto the velocity. Viscous damping is consequently rate dependent: the energydissipation increases with the frequency. In soil dynamics, material dampingis usually assumed to be rate independent in the low frequency range. Rateindependent material damping is sometimes referred to as hysteretic materialdamping [110, 122, 123], although viscous damping also involves a hysteresis effect.

The correspondence principle [173, 178] is often applied to model materialdamping. This principle states that a viscoelastic material can be modelled in thefrequency domain as an equivalent elastic material with modified elastic constants.The application of the correspondence principle results in the use of complex Lamecoefficients (λ + 2µ)(1 + 2βpi) and µ(1 + 2βsi) where βp and βs represent thehysteretic material damping ratio for the dilatational waves and the shear waves,respectively. In the case of rate independent damping, the damping ratios βp

and βs are frequency independent. The use of complex Lame coefficients leadsto complex phase velocities Cp and Cs and complex wavenumbers kp = ω/Cp

and ks = ω/Cs, where ω is the circular frequency. The imaginary part of thewavenumbers corresponds to wave attenuation due to hysteretic material damping.

A wide range of methods to determine the dynamic soil properties is available.A distinction can be made between laboratory tests and in situ tests. Laboratorytests are most often based on the modal analysis of a soil specimen installed in atriaxial cell. These tests include the bender element test [14, 222], the resonantcolumn test [110], the free torsion pendulum test [209], and the cyclic triaxial test[110]. In situ tests are generally based on the measurement of the travel timeand the attenuation of waves travelling over a larger distance. They include theseismic reflection test [122], the seismic refraction test [122, 219], the cross-hole,down-hole, and up-hole test [110], the Seismic Cone Penetration Test (SCPT)[99], and the Spectral Analysis of Surface Waves (SASW) test [151, 231]. In thefollowing, the SASW test and the SCPT are briefly discussed.

Vibration predictions 9

The SASW test

The SASW test is a non-invasive test to determine the dynamic shear modulus ofshallow soil layers [151, 231]. The method is based on the dispersive properties ofsurface waves travelling in a layered halfspace.

The SASW method consists of an in situ experiment to determine the dispersioncurve of the soil and the solution of an inverse problem where the correspondingsoil profile is identified. In the experiment, surface waves are generated by meansof a falling weight device, an impact hammer, or a hydraulic shaker. The free fieldresponse is measured at different receivers up to a distance of typically 50 m. Theexperimental dispersion curve is derived from the phase of the transfer functionsbetween the receivers.

The direct stiffness method [113] or an equivalent formulation is used to calculatethe theoretical dispersion curve of a soil with a given stiffness profile, which isiteratively adjusted in order to minimize the distance between the theoretical andthe experimental dispersion curve. Rix et al. [175] follow a similar approach todetermine the soil’s material damping profile from the attenuation of the surfacewaves.

The seismic cone penetration test

The SCPT is a variant of the classical cone penetration test [99, 110]. In the SCPT,the cone is equipped with two receivers (geophones or accelerometers), verticallyseparated by typically 1 m. A seismic source is installed next to the penetrationpoint of the cone. The source consists of a foundation that is excited by meansof a vertical or a horizontal hammer impact. Dependent on the direction of thehammer impact, P-waves or S-waves propagate through the soil along the shaft.These waves are recorded by the receivers installed in the cone.

The wave velocity is estimated from the arrival times of the waves at bothreceivers or from the phase of the transfer function between both receivers. Thematerial damping ratio of the soil can be estimated from the modulus of thetransfer function, provided that the effect of geometrical attenuation is properlyeliminated. In order to estimate the variation of the soil properties with depth,test is repeated for various cone penetration depths.

1.2.3 Validation of the predictions

Several of the vibration prediction models considered in subsection 1.2.1 have beenvalidated by means of elaborate measurement campaigns. This subsection focuseson the experimental validation of the source models for road and rail traffic inducedvibrations developed by Lombaert et al. [135, 136]. The aim of this subsection isto indicate the motivation of the present thesis, not to give a detailed overviewof the experimental validation. More details on the validation can be found inreferences [134] and [136].

10 Introduction

(a)−0.5 0 0.5 1 1.5 2−3

−2

−1

0

1

2

3x 10

−3

Time [s]

Vel

ocity

[m/s

]

8 m

0 25 50 75 100 125 1500

1

2

3x 10

−4

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

8 m

(b)−0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Time [s]

Vel

ocity

[m/s

]

24 m

0 25 50 75 100 125 1500

0.5

1

1.5x 10

−4

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

24 m

Figure 1.1: Measured (black line) and predicted (gray line) vertical free fieldvelocity at (a) 8 m and (b) 24 m from the center of the road due to the passage ofa truck on an artificial unevenness.

The model for road traffic induced vibrations is validated for a truck passing onan artificial unevenness with the shape of a traffic plateau [134]. The parametersconcerning the vehicle, the road, and the soil are determined experimentally andused to predict the free field vibrations. A two-dimensional vehicle model withfour degrees of freedom is used to calculate the vehicle’s response and the axleloads. The parameters of the model are determined by means of truck weighings,data provided by the truck manufacturer, and a fit of the measured and thepredicted vehicle response. The road is modelled as an infinite elastic beam witha rigid cross section, consisting of an asphalt layer on a crushed stone layer anda crushed concrete layer. The thickness and the Young’s modulus of the asphaltlayer are derived from an SASW test. The other road properties are estimatedfrom guidelines for the design of asphalt roads [156]. The soil is modelled asa layered halfspace. The thickness and the shear wave velocity of the layers arederived from an SASW test, and the dilatational wave velocity from an SCPT. Thedensity is estimated based on the type of soil and the depth of the ground watertable. The hysteretic material damping ratio is estimated from the attenuation ofthe measured free field vibrations.

The free field response is predicted with the source model developed by Lombaert

Vibration predictions 11

et al. [135] and compared with experimental data. Figure 1.1 shows the timehistory and the frequency content of the measured and the predicted vertical freefield velocity at 8 m and 24 m from the center of the road, caused by the passageof a truck on the unevenness at 58 km/h. The time history reveals the passage ofthe front and the rear axle on the unevenness, which are well separated in time.The measured and predicted time histories correspond well. The main differenceis the duration of the transient signal that corresponds to the passage of the rearaxle on the unevenness. This is due a loss of contact between the rear axle andthe road, followed by an impact on the road [134]. This phenomenon is not takeninto account with the linear vehicle model. The frequency content of the free fieldvelocity is dominated by the axle hop frequencies of the vehicle and the verticalresonance of the soil at this site, causing a high response between 10 Hz and 16 Hz.Apart from a misfit around 10 Hz, which is due to the loss of contact between therear axle and the road [134], the correspondence between predicted and measuredresults is very satisfactory. Both the locations and the magnitudes of the maximain the frequency content of the free field response are well predicted.

The model for train induced vibrations developed by Lombaert et al. [136] isvalidated by means of several experiments at the occasion of the homologationtests of the new high speed railway track L2 between Brussels and Koln. The trackunevenness and the dynamic soil and track parameters are determined by meansof in situ experiments. The track unevenness is measured by a track recordingcar and extrapolated to the wavenumber range of interest. The soil is modelledas a layered halfspace. The shear wave velocity and the thickness of the layers aredetermined by means of SASW tests. Values are assumed for the Poisson’s ratio,the density, and the material damping ratio. The dynamic track characteristicsare derived from the known geometry of the track and a fit of the predicted andthe measured rail receptance. The predicted rail receptance depends on the soilprofile determined from the SASW tests. As a result, potential errors in the soilprofile might be compensated for in the identification of the track characteristics.As the influence of the sprung mass of the vehicle on the dynamic axle loads canbe neglected [193], only the unsprung mass is taken into account in the predictionof the free field vibrations.

Figure 1.2 shows the time history and the frequency content of the measured andthe predicted vertical free field velocity at 8 m and 24 m from the track, caused bythe passage of a high speed train at 294 km/h. At 8 m from the track, the passageof individual bogies can be identified in the time history of the response. This isno longer the case at 24 m from the track. In both the predicted and measuredresults, the duration of the response increases in a similar way with the distancefrom the track. The amplitude of the predicted and measured response does notcompletely agree, however. The frequency content of the free field response ismainly situated in the range up to 100 Hz, which is higher than for road trafficinduced vibrations. The modulation of the frequency content is related to thebogie passage frequency fb = 4.4 Hz and the axle passage frequency fa = 27.2 Hz.The correspondence between the frequency content of the predicted and measured

12 Introduction

(a)−3 −2 −1 0 1 2 3 4

−2

−1

0

1

2x 10

−3

Time [s]

Vel

ocity

[m/s

/]

8 m

0 25 50 75 100 125 1500

1

2

3

4

5x 10

−4

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Vel

ocity

[m/s

/Hz]

8 m

(b)−3 −2 −1 0 1 2 3 4

−5

0

5x 10

−4

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ocity

[m/s

/]

24 m

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1x 10

−4

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

24 m

Figure 1.2: Measured (black line) and predicted (gray line) vertical free fieldvelocity at (a) 8 m and (b) 24 m from the track due to the passage of a highspeed train.

response is not as close as for road traffic induced vibrations.In view of the large number of parameters involved, there are several potential

reasons for the lower accuracy of the prediction. The parameters are related to thedynamic behaviour of the vehicle, the track unevenness, the geometry and materialproperties of the track, and the dynamic soil properties. It is also possible thatthe lower accuracy is due to modelling errors, related to the model itself instead ofthe parameters. Possible causes of modelling errors include non-linear constitutivebehaviour of the track or soil, anisotropy of the soil, and a soil profile that deviatesfrom a layered halfspace.

In this work, the focus is on the influence of parametric errors in the soil profile.In both examples considered in this subsection, the vibration predictions are basedon a soil profile determined from an SASW test. The accuracy of the predictionsis considerably higher in the case of road traffic induced vibrations, however. Apossible explanation is that the soil properties exhibit small scale spatial variationsthat can not be detected by the SASW method. For road traffic induced vibrations,the frequency content of the excitation is low and the wavelength of the dominantwaves in the soil is large. These waves do not resolve the small scale variations ofthe soil properties and are therefore not affected by the inaccuracy of the soil profile

Focus of the thesis 13

derived from the SASW test. For rail traffic induced vibrations, the frequencycontent of the excitation is larger and the wavelength of the waves in the soil issmaller. These waves are affected by the small scale variations of the soil propertiesthat are not detected by the SASW method, resulting in inaccurate vibrationpredictions. The results shown in figure 1.2 appear to contradict this hypothesis, asthe correspondence between predicted and measured results does not vary with thefrequency. It is possible, however, that the determination of the other parameters(such as the dynamic track characteristics) has lead to a compensation of errorsin the high frequency range, resulting in a redistribution of error over the entirefrequency range.

1.3 Focus of the thesis

The objective of this thesis is to assess the uncertainty on a soil profile obtainedfrom an SASW test and to study the impact of this uncertainty on the prediction ofground vibrations. However, the methodology elaborated in the thesis is generaland therefore also applicable to other soil investigation techniques, such as theSCPT.

The problem addressed in the thesis is more sharply delineated in subsection1.3.1. The solution strategy is outlined in subsection 1.3.2 and the originalcontributions of the thesis are summarized in subsection 1.3.3.

1.3.1 Problem definition

In the literature, a distinction is made between aleatoric and epistemic uncertainty[62, 149, 227].

Aleatoric or inherent uncertainty refers to an intrinsic variability of the modelledphysical system under consideration. This type of uncertainty is irreducible: evenwhen all information on a property is available, the quantity remains uncertain.Typical examples are dimensions subjected to manufacturing tolerances or thewind pressure on a structure.

Epistemic or subjective uncertainty refers to uncertainty due to the lack ofknowledge. This type of uncertainty is reducible: as more information on aproperty becomes available, the uncertainty decreases and the quantity tends to adeterministic value.

The present work deals with epistemic uncertainty: the aim is to investigate theinfluence of the lack of knowledge on the material properties of an inhomogeneoussoil at a specific site. While it is theoretically possible to acquire all informationneeded for an exact deterministic soil model, this is practically not feasible. Fenton[66] states that the only way to eliminate all uncertainty on the soil properties is toexcavate the entire site. This is clearly unfeasible, and the (remaining) epistemicuncertainty has to be accounted for in ground vibration predictions.

14 Introduction

Whereas aleatoric and epistemic uncertainty are rather different in nature, theyare modelled by the same mathematical formulations. Classically, a probabilisticapproach is followed and the uncertain properties are modelled as random variablesor random processes, characterized by mean values, variances, covariances,probability distributions, and joint probability distributions [119]. The use ofrandom processes allows to model the spatial variation of the uncertain properties.

A Bayesian approach is usually followed to account for the reduction of epistemicuncertainty through measurements [21]. Starting from the available knowledge(from engineering experience or previous experiments), a prior stochastic model isconstructed. The measurement data are used to update this model, leading to aposterior stochastic model with reduced uncertainty.

In the present thesis, a probabilistic model is used to account for the uncertaintyon the soil properties in ground vibration predictions. Ideally, all dynamicsoil properties are modelled as mutually dependent three-dimensional randomprocesses. This is impractical for two reasons: the characterization of sucha stochastic model is not feasible and modelling wave propagation in a fullyheterogeneous three-dimensional medium is computationally very demanding. Thesoil is therefore modelled as a layered halfspace, neglecting the spatial variationof the soil properties in the horizontal direction. Moreover, only the uncertaintyon the dynamic shear modulus is taken into account. The other dynamic soilproperties are modelled as deterministic. A Bayesian approach is followed todetermine the variation of the shear modulus with depth from an SASW test. Theposterior stochastic soil model is used to assess the resolution of the SASW testand the resulting uncertainty in the prediction of ground vibrations.

The uncertainty on the material damping ratio is also important. This isemphasized in the present thesis by means of a brief stochastic study, using ahomogeneous soil model, thus neglecting the spatial variation of the damping ratioin both the horizontal and the vertical direction. The choice to focus primarilyon the dynamic shear modulus is inspired by the availability of well-establishedmeasurement techniques for this soil property, such as the SASW test.

1.3.2 Solution strategy

A dynamic soil-structure interaction experiment is performed on a site in Lincent(Belgium) [187]. Ground-borne vibrations are generated by means of a hammerimpact on a small concrete foundation and recorded with ten accelerometers inthe free field. The transfer functions H(ω) between the (vertical) hammer forcep(ω) and the vertical free field displacements u(ω) are estimated. The experimentis addressed in more detail in section 1.4.

This basic dynamic soil-structure interaction problem is considered as abenchmark problem throughout the thesis. The aim of the thesis is to simulatethe experimental foundation-soil transfer functions, accounting for the uncertaintyon the dynamic shear modulus of the soil.

Focus of the thesis 15

Compared to the dynamic soil-structure interaction problems considered insection 1.2.1, this problem is relatively simple: the foundation is located at thesoil’s surface, behaves as a rigid body, and is excited with an instrumented hammerso that the impact force can directly be measured. However, this basic soil-structure interaction problem comprises the most important elements encounteredin more complicated problems, in particular the transmission of waves throughthe soil. Moreover, the methodology is general and can also be applied tomore complicated problems such as the prediction of road or rail traffic inducedvibrations. The application to a simple problem allows to concentrate on the wavepropagation in the soil.

The dynamic shear modulus at the site in Lincent is determined from anSASW test. The inverse problem in the SASW method is solved by means ofa Bayesian updating scheme. A prior stochastic soil model is first formulated.In the prior model, the variation of the dynamic shear modulus with depth ismodelled as a random process, characterized by a marginal probability distributionand a covariance function. This model represents the knowledge on the shearmodulus that is available before the SASW test is performed. The shear modulusis modelled as a memoryless transformation of a Gaussian process [83]. ThisGaussian process is discretized by means of a Karhunen-Loeve decomposition, ina similar way as in the stochastic finite element method [78]. The prior modelis subsequently updated by means of a Markov chain Monte Carlo method. Thismethod is developed in the field of geophysics [150] and mainly used to investigatethe earth’s crust and upper mantle [131, 192]. The theoretical dispersion curve iscalculated with the direct stiffness method [113] for soil profiles randomly drawnfrom the prior model. Only the soil profiles with a theoretical dispersion curveclose to the experimental curve obtained from the SASW test are accepted. Theresulting ensemble of soil profiles represents the posterior stochastic soil model: itcombines the prior knowledge and the measurement results.

The ensemble of acceptable soil profiles is used in a Monte Carlo simulation topredict the foundation-soil transfer functions. For each realization, a deterministicdynamic soil-structure interaction problem is solved. A method based on thesubdomain formulation developed by Aubry and Clouteau [16, 36] is used. Thisformulation makes use of a finite element model for the foundation and a boundaryelement model for the soil. The boundary element model is based on the Green’sfunctions of a layered halfspace. These functions are calculated in the frequency-wavenumber domain by means of the direct stiffness method [113] and transformedto the frequency-space domain by means of a logarithmic Hankel transformationalgorithm [212]. The results of the Monte Carlo simulation are compared withexperimental data to verify the inversion. As the material damping ratio is notaccounted for in the inversion procedure, a perfect correspondence of predictedand measured data can not be expected, however.

16 Introduction

1.3.3 Original contributions

The calculation of the two-dimensional and three-dimensional Green’s displace-ment and stress functions of a layered soil is treated in detail in the presentwork. While all elements involved in the calculation belong to the state ofthe art since more than 25 years, an attempt is made to combine them in acoherent and comprehensive overview. The overview is inspired by a course texton seismic wave propagation in layered media [46] where the calculation of thetwo-dimensional Green’s displacement functions is addressed. A MATLAB toolbox(ElastoDynamics Toolbox 2.0, EDT 2.0) has been developed to model seismic wavepropagation. Particular attention has been paid to an efficient implementation,as this toolbox is used in (inherently computationally intensive) Monte Carlosimulations to calculate the Green’s functions and the theoretical dispersion curveof soils with uncertain properties.

The Green’s functions are transformed from the wavenumber domain to thespace domain by means of a Hankel transformation algorithm developed by Talman[212]. This algorithm is based on a logarithmic sampling of the wavenumberand spatial coordinates. It is therefore very efficient in the context of problemsinvolving considerably different length scales, such as dynamic soil-structureinteraction problems, where both the interaction between adjacent points on thesoil-structure interface and the radiation of waves to the far field are of interest.The original algorithm is improved through the use of a window and a filterto mitigate artifacts due to the Gibbs phenomenon. Furthermore, guidelinesare formulated for the sampling in the wavenumber domain. The algorithm isintegrated in the MATLAB toolbox EDT 2.0.

Ingredients of the stochastic finite element method [78] are combined with thedirect stiffness method [113] to model wave propagation in a soil with a shearmodulus that varies randomly with depth. This approach allows to account forthe uncertainty on the soil properties in an unbounded domain. In the literatureon stochastic mechanics, Young’s moduli and shear moduli are often modelledas Gaussian variables or processes [32, 78, 82, 109, 140, 211, 229]. While theassumption of a Gaussian distribution leads to numerically tractable formulations,it is physically unrealistic due to the probability that a Gaussian variable takesa negative value. In the present work, a non-Gaussian translation process [83] isused to model the dynamic shear modulus.

The inverse problem in the SASW method is solved by means of a Bayesianupdating technique. The resulting stochastic soil model is used to assess theuncertainty in the prediction of ground vibrations. The predictions are comparedwith experimental data. In an effort to maximize the rate of convergence ofthe inversion procedure, only the components of the shear modulus that havean impact on the predicted ground vibrations are identified.

A basic dynamic soil-structure interaction problem 17

1.4 A basic dynamic soil-structure interaction

problem

The experiment in Lincent is described in this introductory section in order toilluminate the target output of the simulations in the following chapters. Themeasurement site and the experimental configuration are briefly addressed, and theexperimental data are presented. A physical interpretation of the data is omittedhere, but given in the following chapters along with the numerical simulations.

Figure 1.3: Measurement site at the Rue de la Bruyere in Lincent.

The site in Lincent is located next to the Rue de la Bruyere and the high speedrailway track L2 between Brussels and Liege. Figure 1.3 shows a view of the site.Several measurement campaigns have been performed on this site. In preparationof the construction of the high speed railway track, borings and cone penetrationtests have been carried out [110]. Furthermore, SCPTs and SASW tests have beenperformed [110, 167]. The borings revealed the presence of a silt top layer witha thickness of about 1.2 m, followed by a fine sand layer reaching to a depth of3.2 m. Below the shallow layers, a sequence of very stiff layers of arenite and claywas found. The SASW tests indicated the presence of a layer with a thickness

18 Introduction

of 3 m and a shear wave velocity between 150 m/s and 160 m/s on a halfspacewith a shear wave velocity between 250 m/s and 280 m/s. The shear wave velocityresulting from the SCPTs increases almost linearly from about 160 m/s at 1 mdepth to about 280 m/s at 6 m depth.

Figure 1.4: Location of the measurement line.

Figure 1.4 shows a map of the site in Lincent. The location of the foundation andthe accelerometers is indicated. Figure 1.5a shows the foundation and the impacthammer. The square concrete foundation is cast in situ to obtain an optimalcontact with the soil. The width of the foundation is 0.5 m, the height is 0.2 m,and the mass is 125 kg. The impact hammer has a mass of 5.5 kg and is equippedwith a soft tip. The impact force is measured by means of a force sensor built intothe hammer. Figure 1.5b shows one of the accelerometers used to measure the freefield response. The accelerometers are mounted on aluminium stakes with a lengthof 0.3 m. They are located along the measurement line at distances of 2 m, 3 m,4 m, 6 m, 8 m, 12 m, 16 m, 24 m, 32 m, and 48 m from the center of the foundation.

For each receiver, the foundation-soil transfer function H(ω) from the (vertical)hammer force p(ω) to the vertical free field displacement u(ω) is determined fromten impact measurements using the H1 estimator [64]. The displacement u(ω) isobtained from the measured acceleration a(ω) as u(ω) = −a(ω)/ω2. As the resultsare only considered in the frequency domain, it is unnecessary to apply a high-passfilter in the calculation of the displacement u(ω) from the acceleration a(ω). Inaddition to the transfer function H(ω), the coherence function Γ(ω) between theforce p(ω) and the displacement u(ω) is computed [64].

Organization of the text 19

(a) (b)

Figure 1.5: (a) The concrete foundation and the impact hammer and (b) anaccelerometer in the free field.

The results are shown in figure 1.6. A high coherence is observed between20 Hz and 150 Hz, indicating a high signal-to-noise ratio, except at 32 m from thefoundation where the coherence decreases above 100 Hz. In the frequency rangewhere the coherence is high, the measured transfer functions H(ω) are used as areference to verify the numerical predictions in the following chapters.

1.5 Organization of the text

The thesis consists of two parts, focusing on the deterministic and the stochasticprediction of ground-borne vibrations (figure 1.7). The stochastic problem is solvedby means of a Monte Carlo method. A Monte Carlo method involves the solutionof a large sequence of deterministic problems. Much effort has therefore beendevoted to the efficient solution of the deterministic problem. This is reflected bythe detail in which the deterministic problem is addressed in the present text.

Both parts start with one or two chapters on the theoretical background of theproblem, followed by a chapter where the forward problem of vibration predictionsis considered (using a synthetic soil profile), and a chapter on the inverse problemin the SASW method (used to update the soil profile in the forward problem).

Chapter 1 introduces the thesis by situating the subject, highlighting theown contributions and clarifying the organization of the text. In addition,an experiment performed in Lincent is described where ground vibrations are

20 Introduction

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce [

− ]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce [

− ]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce [

− ]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Coh

eren

ce [

− ]

32 m

Figure 1.6: Modulus of the foundation-soil transfer function H(ω) (left hand side)and coherence function Γ(ω) (right hand side) for different receivers, located at(a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation. Themodulus of the transfer function H(ω) is plotted as a solid line if the coherencefunction Γ(ω) exceeds a threshold value Γmin = 0.95.

Organization of the text 21

Deterministic model Stochastic modelT

heo

retica

lback

gro

und

Vib

ration

pre

dic

tions

Soil

chara

cter

ization

2. The direct stiffnessmethod for layered soils

3. Green’s functions oflayered soils

4. Dynamic soil-structureinteraction

5. The SASW method

6. Stochastic mechanics

7. Wave propagation in asoil with uncertainproperties

8. Stochastic soilcharacterization by meansof the SASW method

Figure 1.7: Organization of the text.

generated by a hammer impact on a small foundation. This dynamic soil-structureinteraction problem is considered as a benchmark problem throughout the thesis.

Chapter 2 gives an overview of the direct stiffness method for layered soils [113].The direct stiffness method is used to calculate the wavenumber domain Green’sfunctions of the soil and to determine the theoretical dispersion curve in the SASWmethod.

Chapter 3 treats the transformation of the two-dimensional and three-dimensional Green’s functions from the wavenumber domain to the space domain.Analytical expressions are derived and an efficient numerical algorithm for thetransformations is described [212].

Chapter 4 briefly reviews the subdomain formulation for dynamic soil-structureinteraction developed by Aubry and Clouteau [16, 36]. This method is applied tosimulate the foundation-soil transfer functions measured at the site in Lincent. The

22 Introduction

soil is modelled by means of a boundary element formulation [25, 54], using thethree-dimensional Green’s functions of a layered halfspace discussed in chapter 3.The simulations are based on a synthetic soil profile.

Chapter 5 focuses on the SASW test [151, 231]. The calculation of theexperimental and the theoretical dispersion curve is explained. The methodis applied to determine the dynamic shear modulus of the soil at the site inLincent. The resulting soil profile is used to recalculate the foundation-soil transferfunctions, which are compared with experimental data.

Chapter 6 outlines the Kolmogorov probability theory, which serves as atheoretical framework to define random vectors and random processes. Methodsto model random vectors and random processes are presented. These methodsare used in the stochastic finite element method [78], which is the mostpopular computational method in stochastic mechanics. Three frequently usedformulations of the stochastic finite element method are considered in this chapter:the perturbation stochastic finite element method, the spectral stochastic finiteelement method and the Monte Carlo stochastic finite element method.

Chapter 7 focuses on wave propagation in a soil with uncertain properties.Ingredients of the Monte Carlo stochastic finite element method are applied to solvethe dynamic soil-structure interaction problem considered in chapter 4. First, theimpact of an uncertain shear modulus, Poisson’s ratio, density, and damping ratioon the foundation-soil transfer functions is assessed, assuming that these propertiesdo not vary in space. Second, the effect of the variation of the shear modulus withdepth is investigated. The simulations are performed using a synthetic stochasticsoil model.

Chapter 8 describes the construction of a stochastic soil model accountingfor the uncertainty on the dynamic shear modulus that stems from the SASWtest. A Bayesian approach is followed where prior information is combined withexperimental data obtained from an SASW test. The resulting posterior stochasticsoil model is used to recalculate the stochastic foundation-soil transfer functionsconsidered in chapter 7. The results are compared with the measured transferfunctions.

Chapter 9 summarizes the conclusions of the thesis and gives recommendationsfor further research.

The original contributions of the thesis are mainly situated in chapters 3, 7, and 8.The other chapters reflect the current state of the art and are included to providea coherent framework for the development of the methodology.

Chapter 2

The direct stiffness method

for layered soils

2.1 Introduction

The constitutive behaviour of soil under dynamic loading is complex. Soil isa discontinuous material, where the pores of the solid skeleton can be partlysaturated with water. Laboratory tests show that the soil behaviour is anisotropicand non-linear. For cohesionless dry soils, the non-linear soil behaviour can beneglected when the shear strain γ is smaller than 10−5. This is the case for thevibrations in the free field induced by road and railway traffic. Wave propagation insaturated and unsaturated isotropic poroelastic media can be described by Biot’sporoelastic equations [23, 24]. In the (low) frequency range of interest in mostengineering applications, however, a high viscous coupling prevents the relativemotion between the solid phase and the fluid phase. The soil can be modelled asa dry elastic medium, provided that the density and the incompressibility of thesaturated soil layers are accounted for [185]. Only dry soils are therefore consideredin the present work.

The soil is modelled as a layered elastic halfspace, where the dynamic materialproperties vary only in the vertical direction. The assumption of horizontal soillayers is motivated by the fact that the formation of a soil layer is governed byphenomena affecting large areas of land, such as erosion, sediment transport, andweathering processes [67].

Closed-form solutions for wave propagation in layered media do not exist,necessitating the use of numerical tools [111]. In the present work, the directstiffness method [111, 113] is used to model wave propagation in a layered medium.The direct stiffness method is based on the transfer matrix approach, initiallyproposed by Thomson [213] and Haskell [97], and recast into a stiffness matrixformulation by Kausel and Roesset [113]. The method has also been referred to as

23

24 The direct stiffness method for layered soils

a spectral element formulation by Doyle [58, 59, 60] and Rizzi and Doyle [176, 177].The direct stiffness method is based on a transformation from the time-space

domain to the frequency-wavenumber domain. In the frequency-wavenumberdomain, exact solutions can be obtained for the Navier equations governingwave propagation in a homogeneous layer or a homogeneous halfspace. Thesesolutions are used to formulate element stiffness matrices for homogeneous layerand halfspace elements. Element stiffness matrices express the relation betweenthe displacements and tractions on the boundaries of an element. The stiffnessmatrix of a layered soil is obtained from the assembly of element stiffness matrices.The direct stiffness method can be regarded as a special form of the finite elementmethod, using exact solutions as shape functions. Due to the use of these specificshape functions, wave propagation is treated exactly and there is no need tosubdivide homogeneous layers into multiple layer elements.

As an alternative to the direct stiffness method, the thin layer method can beused [113]. The thin layer method is based on the use of polynomial shape functionsto represent the vertical variation of displacements and tractions. Comparedto the direct stiffness method, the thin layer method leads to mathematicallymore tractable stiffness matrices involving only polynomial functions instead oftranscendental functions. Due to its approximative nature, the thin layer methodrequires a small thickness of the layer elements compared to the smallest relevantwavelength. Furthermore, the method is only applicable to a layered soil supportedby a rigid stratum. A hybrid formulation, where thin layer elements are coupled toa halfspace element, offers a solution, but again leads to transcendental functionsin the stiffness matrix.

The direct stiffness method can be used to solve a wide variety of problems,including amplification of waves in layered media, the computation of dispersivewave modes in layered media, and the computation of the forced response of layeredmedia due to harmonic or transient loading. In the present work, it is used tocalculate the Green’s functions and the theoretical dispersion curve of a layeredsoil. The Green’s functions are needed to solve dynamic soil-structure interactionproblems by means of the subdomain formulation addressed in chapter 4. Thetheoretical dispersion curve is used to determine the soil’s dynamic shear modulusin the SASW method, which is considered in chapter 5.

While the direct stiffness method belongs to the state of the art since the early1980s, it is treated in detail in this chapter. The aim is to provide a soundframework for the calculation of the Green’s functions and the dispersion curveof a layered soil, using a consistent notation. The chapter is inspired by a coursetext on seismic wave propagation in layered media [46]. Compared to the coursetext, the calculation of strains and stresses is further elaborated here, as well asthe calculation of the response at locations within elements. In the frame of thepresent work, a MATLAB toolbox (ElastoDynamics Toolbox 2.0, EDT 2.0) isdeveloped, containing all elements discussed in this chapter.

The chapter is organized as follows. Section 2.2 addresses the elastodynamicequations governing wave propagation in a homogeneous medium. The dis-

Governing equations 25

placement vector is decomposed in terms of three wave potentials that describedilatational waves, vertically polarized shear waves, and horizontally polarizedshear waves. The wave potentials are governed by three uncoupled partialdifferential equations. In section 2.3, two-dimensional wave propagation in alayered medium is considered. The equations for the wave potentials are solvedby means of a transformation from the time-space domain to the frequency-wavenumber domain. Expressions are derived that relate the displacementsand tractions on a horizontal plane to the wave potentials. In section 2.4, thethree-dimensional case is addressed in a similar way. Section 2.5 focuses onthe element stiffness matrices and shape functions used in the direct stiffnessmethod. They follow from the elimination of the wave potentials from theequations derived in sections 2.3 and 2.4. The element stiffness matrices areassembled to obtain the stiffness matrix of a layered soil. In a similar way as in thefinite element method, the displacements at the interfaces between elements arecalculated from the external tractions using the stiffness matrix. Displacementsand tractions at locations within elements are subsequently obtained from theinterface displacements using the shape functions. The application of the directstiffness method to calculate the two-dimensional and three-dimensional Green’sfunctions of a layered soil is addressed in the next chapter. The calculation of thetheoretical dispersion curve of a layered soil is treated in chapter 5.

2.2 Governing equations

In this section, the elastodynamic equations are addressed. Wave potentials areintroduced to obtain uncoupled partial differential equations governing dilatationaland shear wave propagation in a homogeneous medium. The equations areformulated in a Cartesian and a cylindrical frame of reference. The Cartesianframe of reference is used in section 2.3 to solve the elastodynamic equations fortwo-dimensional wave propagation. The cylindrical frame of reference is used insection 2.4 for the three-dimensional case.

Throughout the thesis, a right-handed Cartesian or cylindrical frame of referenceis used, with the origin at the soil’s surface and the z-axis pointing downwards.

2.2.1 Cartesian coordinates

The elastodynamic equations

In a Cartesian frame of reference, the components of the displacement vector inan elastic medium at a position x and a time t are denoted as ui(x, t). Thecomponents εij of the small strain tensor are related to the displacements by thefollowing linearized strain-displacement relations:

εij =1

2(uj,i + ui,j) (2.1)

26 The direct stiffness method for layered soils

Herein, uj,i denotes the derivative of uj with respect to the i-th spatial coordinate.For a Green elastic material, Hooke’s law relates the Cauchy stress tensor σij to

the small strain tensor εkl:

σij = Cijklεkl (2.2)

where the summation convention is used. In equation (2.2), Cijkl is the elasticitytensor. For an isotropic material, this tensor is given by:

Cijkl = λδijδkl + µ(δikδjl + δilδjk) (2.3)

where δij is the Kronecker Delta and λ and µ are the Lame constants. Theseconstants are related to the Young’s modulus E and the Poisson’s ratio ν:

λ =Eν

(1 + ν)(1 − 2ν)(2.4)

µ =E

2(1 + ν)(2.5)

Using equations (2.2) and (2.3), the constitutive equations become:

σij = λεkkδij + 2µεij (2.6)

The dynamic equilibrium of the elastic medium is expressed as:

σji,j + ρbi = ρui (2.7)

where ρbi are the body forces and ρ is the density. A dot above a variable denotesdifferentiation with respect to time.

The Navier equations result from the introduction of the constitutive equations(2.6) and the strain-displacement relations (2.1) in the equilibrium equations (2.7):

(λ+ µ)uj,ij + µui,jj + ρbi = ρui (2.8)

In vector notation, the Navier equation is formulated as:

(λ+ µ)∇∇ · u + µ∇2u + ρb = ρu (2.9)

The operator ∇ is defined as ∂/∂x, ∂/∂y, ∂/∂zT. Hence, ∇2 = ∂2/∂x2 +∂2/∂y2 + ∂2/∂z2 is the Laplace operator and ∇u, ∇ · u, and ∇ × u denote thegradient, the divergence, and the curl of u. Using the following relation:

∇2u = ∇∇ · u −∇×∇× u (2.10)

the Navier equation can alternatively be written as:

(λ+ 2µ)∇∇ · u− µ∇×∇× u + ρb = ρu (2.11)

Governing equations 27

The solution of the Navier equation (2.11) has to satisfy the initial and boundaryconditions of the elastodynamic problem. The following initial conditions areformulated for the displacements u(x, t) and the velocities u(x, t) at the timet = 0:

u(x, 0) = u0(x) (2.12)

u(x, 0) = u0(x) (2.13)

The boundary Γ of the domain Ω with unit outward normal vector n can besubdivided into two parts Γu and Γt where the displacements and the surfacetractions are prescribed:

u(x, t) = u(x, t) on Γu (2.14)

tn(x, t) = tn(x, t) on Γt (2.15)

The tractions tn on a plane with unit outward normal vector n are obtainedaccording to Cauchy’s stress principle:

tni = σijnj (2.16)

In the case where the domain Ω is unbounded, additional restrictions are imposedon the behaviour of the field variables at infinity. These restrictions are calledSommerfeld’s radiation conditions [63]. Consider a sphere ΩR with a limitinglarge radius R and a boundary ΓR with unit outward normal vector n. Thedisplacements u on the boundary ΓR of the domain ΩR are decomposed as u =un+us, where un = u·n and us = u−un are the components normal and tangentialto the boundary. Similarly, the tractions tn on the boundary are decomposed astn = tnn + tns , where tnn = tn · n and tns = tn − tnn are the components normaland tangential to the boundary. The magnitudes of the vectors un, us, tnn , and tnsare denoted by un, us, t

nn , and tns , respectively. The following equations represent

Sommerfeld’s radiation conditions:

tnn + ρCpun = o

(

1

R

)

(2.17)

tns + ρCsus = o

(

1

R

)

(2.18)

where Cp =√

(λ+ 2µ)/ρ and Cs =√

µ/ρ denote the shear and dilatational wavevelocity, respectively. It can be shown that Sommerfeld’s radiation conditionsimply that only an outward flow of energy is allowed at infinity [63].

Helmholtz decomposition of the displacement vector

In the following, the body forces ρb are not accounted for and the homogeneousNavier equation is used:

(λ+ 2µ)∇∇ · u− µ∇×∇× u = ρu (2.19)

28 The direct stiffness method for layered soils

In classical elastodynamics [2], it is customary to explain the physical meaning ofthe Navier equation by a Helmholtz decomposition of the displacement vector intotwo parts: the first component is the gradient of a scalar function Φ, while thesecond component is the curl of a vector function Ψ:

u = ∇Φ + ∇× Ψ (2.20)

As the three displacement components are written in terms of four scalar potentialfunctions Φ, Ψx, Ψy, and Ψz, an additional relation must hold. According toAchenbach [3], the vector Ψ satisfies:

∇ · Ψ = 0 (2.21)

Using the Helmholtz decomposition (2.20), the homogeneous Navier equation(2.19) is transformed in the following set of uncoupled partial differential equations:

(λ+ 2µ)∇2Φ = ρΦ (2.22)

µ∇2Ψ = ρΨ (2.23)

The dilatational motion, described by the scalar potential Φ, uncouples from therotational part of the disturbance, described by the vector potential Ψ. Theuncoupling of dilatational and shear waves only occurs in a homogeneous mediumwhen the influence of body forces is neglected. In a layered medium, couplingoccurs at the interfaces between homogeneous layers.

(a) (b)

Figure 2.1: (a) Dilatational and (b) shear wave.

In the dilatational (or: longitudinal, irrotational, primary, P) wave, theparticles move parallel to the wave propagation direction (figure 2.1a). Thedilatational wave velocity equals Cp =

√

(λ+ 2µ)/ρ. In the shear (or: transverse,equivoluminal, rotational, secondary, S) wave, the particles move perpendicularto the wave propagation direction (figure 2.1b). The shear wave velocity is equalto Cs =

√

µ/ρ. The ratio s of the body wave velocities Cs and Cp is equal to√

(1 − 2ν)/(2 − 2ν) and only depends on Poisson’s ratio ν.

Governing equations 29

Vertically and horizontally polarized shear waves

The contribution of the shear wave to the displacement vector can be furtherdecomposed into a component in a plane parallel (SH-wave) and a componentin a plane normal (SV-wave) to a bounding surface. As we are interestedin the response of a horizontally layered medium where the vertical z-axis isperpendicular to the layer interfaces, it is convenient to choose the boundingsurface parallel with the horizontal (x, y)-plane. According to Pilant [163], thefollowing alternative decomposition of the displacement vector u is used:

u = ∇Φ − l∇×∇× (ezΨ′) + ∇× (ezχ) (2.24)

where the scalar potentials Ψ′ and χ describe the propagation of the SV-wave andSH-wave, respectively. The dimensional factor l makes the scalar potentials Φ, Ψ′

and χ of the same dimension.Introducing the decomposition (2.24) into the Navier equation (2.19) results into

the following set of uncoupled partial differential equations:

(λ+ 2µ)∇2Φ = ρΦ (2.25)

µ∇2Ψ′ = ρΨ′ (2.26)

µ∇2χ = ρχ (2.27)

denoting P-wave, SV-wave, and SH-wave propagation, respectively.

In-plane and out-of-plane motion

For two-dimensional wave propagation in the (x, z)-plane, the dependence on they-coordinate vanishes. We therefore choose to work with a scalar potential Ψinstead of Ψ′, defined as follows:

Ψ = l∂Ψ′

∂x(2.28)

The decomposition of the displacement vector u now reduces to:

u = ∇Φ + ∇× (eyΨ) + ∇× (ezχ) (2.29)

or:

ux =∂Φ

∂x− ∂Ψ

∂z(2.30)

uy = −∂χ∂x

(2.31)

uz =∂Φ

∂z+∂Ψ

∂x(2.32)

30 The direct stiffness method for layered soils

The in-plane motions ux and uz uncouple from the out-of-plane motion uy. Thein-plane motion is described in terms of the scalar wave potentials Φ (P-wave) andΨ (SV-wave), while the out-of-plane motion is described in terms of scalar wavepotential χ (SH-wave).

Introducing the decomposition (2.29) in the Navier equation (2.19), the followingset of uncoupled partial differential equations is obtained:

(λ+ 2µ)∇2Φ = ρΦ (2.33)

µ∇2Ψ = ρΨ (2.34)

µ∇2χ = ρχ (2.35)

where the Laplace operator ∇2 reduces to ∂2/∂x2 + ∂2/∂z2.

2.2.2 Cylindrical coordinates

The elastodynamic equations

The elastodynamic behaviour of a medium can also be represented in cylindricalcoordinates [3]. The independent coordinates are (r, θ, z, t), with r and θ the radialand circumferential coordinate, respectively.

The linear strain-displacement relations are:

εrr =∂ur∂r

(2.36)

εθθ =urr

+1

r

∂uθ∂θ

(2.37)

εzz =∂uz∂z

(2.38)

εrθ =1

2

(

∂uθ∂r

− uθr

+1

r

∂ur∂θ

)

(2.39)

εθz =1

2

(

1

r

∂uz∂θ

+∂uθ∂z

)

(2.40)

εzr =1

2

(

∂ur∂z

+∂uz∂r

)

(2.41)

The constitutive equations are:

σrr = λεkk + 2µεrr (2.42)

σθθ = λεkk + 2µεθθ (2.43)

σzz = λεkk + 2µεzz (2.44)

Governing equations 31

σrθ = 2µεrθ (2.45)

σθz = 2µεθz (2.46)

σzr = 2µεzr (2.47)

The equilibrium equations are:

∂σrr∂r

+1

r

∂σrθ∂θ

+∂σzr∂z

+σrr − σθθ

r+ ρbr = ρ

∂2ur∂t2

(2.48)

∂σθr∂r

+1

r

∂σθθ∂θ

+∂σθz∂z

+2σθrr

+ ρbθ = ρ∂2uθ∂t2

(2.49)

∂σzr∂r

+1

r

∂σzθ∂θ

+∂σzz∂z

+σzrr

+ ρbz = ρ∂2uz∂t2

(2.50)

Introduction of the constitutive equations and the strain-displacement relations inthe equilibrium equations leads to the Navier equations.

Helmholtz decomposition of the displacement vector

The displacement vector is decomposed into the gradient of a scalar wave potentialΦ and the curl of a vector potential Ψ = Ψr,Ψθ,ΨzT. The vector potential Ψ

satisfies:

∇ · Ψ = 0 (2.51)

This leads to the following set of hyperbolic partial differential equations:

(λ+ 2µ)∇2Φ = ρΦ (2.52)

µ(∇2Ψr −Ψr

r2− 2

r2∂Ψθ

∂θ) = ρΨr (2.53)

µ(∇2Ψθ −Ψθ

r2− 2

r2∂Ψr

∂θ) = ρΨθ (2.54)

µ∇2Ψz = ρΨz (2.55)

with ∇2 the Laplace operator:

∇2 =∂2

∂r2+

1

r

∂

∂r+

1

r2∂2

∂θ2+

∂2

∂z2(2.56)

No uncoupling of the potentials Ψr and Ψθ is obtained in equations (2.53) and(2.54).

32 The direct stiffness method for layered soils

Vertically and horizontally polarized shear waves

In order to further decompose the shear wave in an SV-wave and an SH-wave, thefollowing alternative decomposition of the displacement vector is used [163]:

u = ∇Φ − l∇×∇× (ezΨ′) + ∇× (ezχ) (2.57)

or:

ur =∂Φ

∂r− l

∂2Ψ′

∂r∂z+

1

r

∂χ

∂θ(2.58)

uθ =1

r

∂Φ

∂θ− l

1

r

∂2Ψ′

∂θ∂z− ∂χ

∂r(2.59)

uz =∂Φ

∂z+ l(∇2Ψ′ − ∂2Ψ′

∂z2) (2.60)

The scalar wave potentials Φ, Ψ′, and χ describe the propagation of the P-waves, the SV-waves, and the SH-waves. The dimensional factor l makes thescalar potentials Φ, Ψ′ and χ of the same dimension. Note that both shear wavescontribute to the radial and circumferential displacements ur and uθ. The resultingset of uncoupled partial differential equations becomes:

(λ+ 2µ)∇2Φ = ρΦ (2.61)

µ∇2Ψ′ = ρΨ′ (2.62)

µ∇2χ = ρχ (2.63)

In section 2.4, the equilibrium equations (2.61–2.63) are solved for cases wherethe displacement field u is symmetric or antisymmetric with respect to the planeθ = 0. If the displacement field u is symmetric, ur and uz are even functions of θand uθ is an odd function of θ. Equations (2.58–2.60) show that the correspondingP-wave potential Φ and SV-wave potential Ψ′ are even functions of θ, while thecorresponding SH-wave potential χ is an odd function of θ. If the displacementfield u is antisymmetric, the P-wave potential Φ and the SV-wave potential Ψ′ areodd functions of θ and the SH-wave potential χ is an even function of θ.

2.3 Two-dimensional wave propagation

In this section, the elastodynamic equations are solved for the case of two-dimensional wave propagation in a horizontally layered medium. First, thesolution (with unknown integration constants) is formulated in terms of the wavepotentials in the frequency-wavenumber domain. Next, the relations between thewave potentials in the frequency-wavenumber domain and the displacements and

Two-dimensional wave propagation 33

tractions (on a horizontal plane) in the frequency-space domain are derived. Theserelations are used to specify boundary conditions for the wave potentials, so thatthe integration constants can be determined. The same relations allow to derivethe unknown displacements and tractions from the wave potentials. Finally, it isshown how strains and stresses are determined from displacements and tractions.

2.3.1 Wave potentials

In the case of two-dimensional wave propagation in the (x, z)-plane, theequilibrium in terms of the wave potentials is given by the partial differentialequations (2.33–2.35). These equations are solved in the frequency-wavenumberdomain by means of a forward Fourier transformation from the time t to thecircular frequency ω, followed by a forward Fourier transformation from thehorizontal coordinate x to the horizontal wavenumber kx.

The relation between the time domain representation f(t) and the frequency

domain representation f(ω) of a function f is given by:

f(ω) = F [f(t);ω] =

∫ ∞

−∞

e−iωtf(t) dt (2.64)

f(t) = F−1[

f(ω); t]

=1

2π

∫ ∞

−∞

e+iωtf(ω) dω (2.65)

where F and F−1 denote the forward and inverse Fourier transformation,respectively.

The relation between the space domain representation f(x) and the wavenumberdomain representation f(kx) of a function f is slightly different:

f(kx) = F [f(x); kx] =

∫ ∞

−∞

e+ikxxf(x) dx (2.66)

f(x) = F−1[

f(kx);x]

=1

2π

∫ ∞

−∞

e−ikxxf(kx) dkx (2.67)

Dilatational waves

In the hyperbolic partial differential equation (2.33) that governs the in-planepropagation of dilatational waves, the time t is transformed to the circularfrequency ω by means of a forward Fourier transformation:

(λ+ 2µ)

[

∂2

∂x2+

∂2

∂z2

]

Φ + ω2ρΦ = 0 (2.68)

where a hat above a variable denotes its representation in the frequency-spacedomain. The correspondence principle [173, 178] is applied in equation (2.68) to

34 The direct stiffness method for layered soils

model material damping in the elastic medium by the use of a complex Lamecoefficient (λ+2µ)(1+2βpi), where βp represents the hysteretic material dampingratio for the dilatational waves.

The previous transformation is followed by a forward Fourier transformation ofthe horizontal coordinate x to the horizontal wavenumber kx. As it is assumed thatthe geometry is invariant in the horizontal direction, this leads to the followingordinary differential equation:

(λ+ 2µ)

[

−k2x +

d2

dz2

]

Φ + ω2ρΦ = 0 (2.69)

where a tilde above a variable denotes its representation in the frequency-wavenumber domain. As the differential equation (2.69) has constant coefficients,the following solution can be proposed:

Φ(kx, z, ω) = P e−ikzpz (2.70)

where kzp represents the vertical component of the wave propagation vector kp =

kx, kzpTof the dilatational waves. The eigenvector component P depends on the

boundary and initial conditions of the problem under consideration. Introducingthe solution (2.70) in the ordinary differential equation (2.69) leads to:

[

−(λ+ 2µ)(k2x + k2

zp) + ω2ρ]

P = 0 (2.71)

A non-trivial solution for the eigenvector component P can be found if thecoefficient of P in equation (2.71) is equal to zero or:

k2x + k2

zp =ω2

C2p

= k2p (2.72)

where Cp =√

(λ+ 2µ)/ρ is the dilatational wave velocity and kp = ω/Cp

is the magnitude of the wave propagation vector kp. The application of thecorrespondence principle [173, 178] in equation (2.68) results in a complex wavevelocity Cp and a complex wavenumber kp. Equation (2.72) is referred to as thedispersion relation for the dilatational wave and allows to calculate the complexvertical wavenumber kzp for each horizontal wavenumber kx and each circularfrequency ω:

kzp = ±√

k2p − k2

x (2.73)

A real value of kzp corresponds to a wave propagating in the vertical direction ez(figure 2.2a), while an imaginary value represents an evanescent or inhomogeneouswave with an exponential decrease or increase in amplitude with z (figure 2.2b).Selecting the sign in equation (2.73) so that Im(kzp) ≤ 0 if kzp is complex and

Two-dimensional wave propagation 35

(a) (b)

Figure 2.2: (a) Vertically propagating and (b) vertically evanescent P-wave.

ωkzp ≥ 0 if kzp is real, the solution (2.70) is written more generally as:

Φ(kx, z, ω) = IPe−ikzpz + RPe

+ikzpz (2.74)

The amplitudes IP and RP refer to incident (outgoing, propagating in the positivez-direction) and reflected (incoming, propagating in the negative z-direction) P-waves, respectively.

The wave potential Φ(x, z, ω) in the frequency-space domain is obtained bymeans of an inverse Fourier transformation of the wavenumber kx to the spatialcoordinate x:

Φ(x, z, ω) =1

2π

∫ ∞

−∞

e−ikxx(

IPe−ikzpz + RPe

+ikzpz)

dkx (2.75)

An additional inverse Fourier transformation of the circular frequency ω to thetime t gives the wave potential Φ(x, z, t) in the time-space domain:

Φ(x, z, t) =1

(2π)2

∫ ∞

−∞

∫ ∞

−∞

ei(ωt−kxx)(

IPe−ikzpz + RPe

+ikzpz)

dkx dω (2.76)

Vertically polarized shear waves

In a similar way as for the dilatational waves, the partial differential equation(2.34) that governs the in-plane propagation of shear waves is transformed to thefrequency-wavenumber domain:

µ

[

−k2x +

d2

dz2

]

Ψ + ω2ρΨ = 0 (2.77)

The correspondence principle is applied to model material damping by the useof a complex Lame coefficient µ(1 + 2βsi), where βs represents the hystereticmaterial damping ratio for the shear waves. The following solution of the ordinarydifferential equation (2.77) can be proposed:

Ψ(kx, z, ω) = Se−ikzsz (2.78)

36 The direct stiffness method for layered soils

where the vertical wavenumber kzs follows from the dispersion relation for theshear wave:

k2x + k2

zs =ω2

C2s

= k2s (2.79)

Herein, Cs =√

µ/ρ is the shear wave velocity and ks = ω/Cs is the magnitude ofthe wave propagation vector ks. Both ks and Cs are complex due to the applicationof the correspondence principle. The dispersion relation (2.79) allows to calculatethe complex vertical wavenumber kzs for each horizontal wavenumber kx and eachcircular frequency ω:

kzs = ±√

k2s − k2

x (2.80)

Selecting the sign in equation (2.80) so that Im(kzs) ≤ 0 if kzs is complex andωkzs ≥ 0 if kzs is real, the solution (2.78) is written more generally as:

Ψ(kx, z, ω) = ISVe−ikzsz + RSVe

+ikzsz (2.81)

The amplitudes IP and RP refer to incident (outgoing, propagating in the positivez-direction) and reflected (incoming, propagating in the negative z-direction) P-waves, respectively.

The wave potential Ψ(x, z, ω) in the frequency-space domain is obtained bymeans of an inverse Fourier transformation of the wavenumber kx to the spatialcoordinate x:

Ψ(x, z, ω) =1

2π

∫ ∞

−∞

e−ikxx(

ISVe−ikzsz + RSVe

+ikzsz)

dkx (2.82)

An additional inverse Fourier transformation of the circular frequency ω to thetime t gives the wave potential Ψ(x, z, t) in the time-space domain:

Ψ(x, z, t) =1

(2π)2

∫ ∞

−∞

∫ ∞

−∞

ei(ωt−kxx)(

ISVe−ikzsz+ RSVe

+ikzsz)

dkx dω (2.83)

Horizontally polarized shear waves

The out-of-plane propagation of shear waves is governed by the partial differentialequation (2.35). This equation is similar to the partial differential equation (2.34)for the in-plane propagation of shear waves and a similar solution is obtained:

χ(kx, z, ω) = ISHe−ikzsz + RSHe

+ikzsz (2.84)

where the vertical wavenumber kzs is defined in the same way as for the in-planepropagation of shear waves.

Two-dimensional wave propagation 37

The wave potential χ(x, z, ω) in the frequency-space domain is obtained bymeans of an inverse Fourier transformation of the wavenumber kx to the spatialcoordinate x:

χ(x, z, ω) =1

2π

∫ ∞

−∞

e−ikxx(

ISHe−ikzsz + RSHe

+ikzsz)

dkx (2.85)

An additional inverse Fourier transformation of the circular frequency ω to thetime t leads to the wave potential χ(x, z, t) in the time-space domain:

χ(x, z, t) =1

(2π)2

∫ ∞

−∞

∫ ∞

−∞

ei(ωt−kxx)(

ISHe−ikzsz + RSHe

+ikzsz)

dkx dω (2.86)

2.3.2 Displacements

Using equations (2.30–2.32), the displacements u can be derived from the wavepotentials Φ, Ψ, and χ in the time-space domain. Alternatively, the displacementsu can be obtained from the wave potentials Φ, Ψ, and χ in the frequency-spacedomain. The time-space domain displacements u are then computed from u bymeans of an inverse Fourier transformation of the circular frequency ω to the timet. In the present work, the second approach is followed and u is calculated in thefrequency-space domain using equations (2.30–2.32):

uxuyuz

=

∂∂x − ∂

∂z 00 0 − ∂

∂x∂∂z

∂∂x 0

Φ

Ψχ

(2.87)

Equation (2.87) clearly shows that the in-plane motion and the out-of-planemotion are uncoupled. The in-plane motion only depends on the wave potentialsΦ and Ψ, while the out-of-plane motion only depends on the wave potentialχ. As a consequence, the relation between the displacements u and the wavepotentials Φ, Ψ, and χ can alternatively be expressed by means of two independentequations governing in-plane wave propagation and out-of-plane wave propagation,respectively. Both are treated simultaneously here in order to limit the number ofequations.

Introducing equations (2.75), (2.82), and (2.85) in equation (2.87) andelaborating the partial derivatives with respect to the spatial coordinates leadsto:

u =1

2π

∫ ∞

−∞

e−ikxx(

BIZIaI + BRZRaR)

dkx (2.88)

Herein, the matrices BI and BR are defined as:

BI =

−ikx ikzs 00 0 ikx

−ikzp −ikx 0

BR =

−ikx −ikzs 00 0 ikxikzp −ikx 0

(2.89)

38 The direct stiffness method for layered soils

The matrices ZI and ZR are defined as:

ZI =

e−ikzpz 0 00 e−ikzsz 00 0 e−ikzsz

ZR =

e+ikzpz 0 00 e+ikzsz 00 0 e+ikzsz

(2.90)

The vectors aI and aR collect the wave potential amplitudes:

aI =

IPISV

ISH

aR =

RP

RSV

RSH

(2.91)

In equation (2.88), the spatial coordinate x is transformed to the horizontalwavenumber kx by means of a forward Fourier transformation:

u = BIZIaI + BRZRaR (2.92)

While it is possible to formulate the equilibrium equations in terms of thefrequency-wavenumber domain displacement vector u, this would lead to anasymmetric stiffness matrix [111]. In order to obtain a symmetric stiffness matrix,the equilibrium equations are formulated in terms of a modified displacementvector u, defined as:

u = Tu (2.93)

where:

T =

1 0 00 1 00 0 i

(2.94)

The modified displacement vector u in the frequency-wavenumber domain isrelated to the displacement vector u in the frequency-space domain as:

u = T

∫ ∞

−∞

eikxxu dx (2.95)

u =1

2πT−1

∫ ∞

−∞

e−ikxxu dkx (2.96)

Using equations (2.92) and (2.93), the modified displacement vector u is expressedas a function of the wave potentials:

u = DIZIaI + DRZRaR (2.97)

where the matrices DI = TBI and DR = TBR are given by:

DI =

−ikx ikzs 00 0 ikxkzp kx 0

DR =

−ikx −ikzs 00 0 ikx

−kzp kx 0

(2.98)

Two-dimensional wave propagation 39

2.3.3 Tractions

The tractions tez = σzx, σzy , σzzT on a horizontal plane with unit outwardnormal vector ez are expressed as a function of the displacements u in thefrequency-space domain using the strain-displacement relations (2.1) and theconstitutive equations (2.6):

tezx

tezy

tezz

=

0 0 0 0 0 2µ0 0 0 0 2µ 0λ λ λ+ 2µ 0 0 0

∂∂x 0 00 ∂

∂y 0

0 0 ∂∂z

12∂∂y

12∂∂x 0

0 12∂∂z

12∂∂y

12∂∂z 0 1

2∂∂x

uxuyuz

(2.99)

Introducing equation (2.88) and elaborating the partial derivatives with respectto the spatial coordinates leads to:

tez =1

2π

∫ ∞

−∞

e−ikxx(

CIZIaI + CRZRaR)

dkx (2.100)

where the matrices CI and CR are given by:

CI =

−2µkxkzp −µk2x + µk2

zs 00 0 µkxkzs

−λk2x − (λ+ 2µ)k2

zp −2µkxkzs 0

(2.101)

CR =

2µkxkzp −µk2x + µk2

zs 00 0 −µkxkzs

−λk2x − (λ+ 2µ)k2

zp 2µkxkzs 0

(2.102)

In equation (2.100), the spatial coordinate x is transformed to the horizontalwavenumber kx by means of a forward Fourier transformation:

tez = CIZIaI + CRZRaR (2.103)

In a similar way as for the displacements, a modified traction vector tez

isintroduced:

tez

= Ttez (2.104)

The modified traction vector tez

in the frequency-wavenumber domain is relatedto the traction vector tez in the frequency-space domain as:

tez

= T

∫ ∞

−∞

eikxxtez dx (2.105)

tez =1

2πT−1

∫ ∞

−∞

e−ikxxtezdkx (2.106)

40 The direct stiffness method for layered soils

Using equations (2.103) and (2.104), the modified traction vector tez

is expressedas a function of the wave potentials:

tez

= GIZIaI + GRZRaR (2.107)

where the matrices GI = TCI and GR = TCR are given by:

GI =

−2µkxkzp −µk2x + µk2

zs 00 0 µkxkzs

−iλk2x − i(λ+ 2µ)k2

zp −2iµkxkzs 0

(2.108)

GR =

2µkxkzp −µk2x + µk2

zs 00 0 −µkxkzs

−iλk2x − i(λ+ 2µ)k2

zp 2iµkxkzs 0

(2.109)

2.3.4 Strains and stresses

The strain components εxx, εyy, and εxy are related to the displacements u throughequation (2.1):

εxxεyyεxy

=

∂∂x 0 00 ∂

∂y 012∂∂y

12∂∂x 0

uxuyuz

(2.110)

Equation (2.110) only involves derivatives with respect to the horizontal coor-dinates. Using equation (2.96), these derivatives are elaborated in terms of themodified displacement vector u without the use of the wave potentials aI and aR:

εxxεyyεxy

=1

2π

∫ ∞

−∞

e−ikxx

−ikx 0 00 0 00 − i

2kx 0

ux

uy

uz

dx (2.111)

The remaining strain components εzx, εzy, εzz and stress components σxx, σyy, σxyare calculated in the space domain from the strain components given by equation(2.111) and the stress components given by equation (2.105) using the constitutiveequations (2.6).

2.4 Three-dimensional wave propagation

In this section, the elastodynamic equations are solved for the case of three-dimensional wave propagation in a horizontally layered medium. The solutionis elaborated along the same lines as for the two-dimensional case. First, the wavepotentials are calculated in the frequency-wavenumber domain. Next, the relationsbetween the wave potentials and the displacements, tractions, strains, and stressesare derived.

Three-dimensional wave propagation 41

2.4.1 Wave potentials

In the case of three-dimensional wave propagation, the equilibrium in terms ofthe wave potentials is given in cylindrical coordinates by the partial differentialequations (2.61–2.63). These equations are solved in the frequency-wavenumberdomain by means of a Fourier transformation from the time t to the circularfrequency ω, followed by a Fourier series expansion from the circumferentialcoordinate θ to the circumferential wavenumber n and a Hankel transformationfrom the radial coordinate r to the radial wavenumber kr.

The Fourier series expansion of a function f(θ) is defined as:

f(θ) =

∞∑

n=0

fc(n) cos(nθ) + fs(n) sin(nθ) (2.112)

The Fourier coefficients fc(n) and fs(n) are given by:

fc(n) = an

∫ 2π

0

f(θ) cos(nθ) dθ (2.113)

fs(n) = an

∫ 2π

0

f(θ) sin(nθ) dθ (2.114)

where the coefficient an is equal to 1/(2π) if n = 0 and 1/π if n 6= 0. If the functionf(θ) is even, the Fourier coefficients fs(n) vanish and equation (2.112) reduces to aFourier cosine series expansion. If the function f(θ) is odd, the Fourier coefficientsfc(n) vanish and equation (2.112) reduces to a Fourier sine series expansion.

The relation between the space domain representation f(r) and the wavenumberdomain representation f(kr) of a function f is given by:

f(kr) = Hn [f(r); kr ] =

∫ ∞

0

rJn(krr)f(r) dr (2.115)

f(r) = H−1n [f(kr); r] =

∫ ∞

0

krJn(krr)f(kr) dkr (2.116)

where Jn is an n-th order Bessel function of the first kind and Hn and H−1n denote

the forward and inverse Hankel transformation of order n, respectively.

Dilatational waves

In the hyperbolic partial differential equation (2.61) that governs the propagationof dilatational waves, the time t is transformed to the circular frequency ω bymeans of a forward Fourier transformation:

(λ+ 2µ)

[

∂2

∂r2+

1

r

∂

∂r+

1

r2∂2

∂θ2+

∂2

∂z2

]

Φ + ω2ρΦ = 0 (2.117)

42 The direct stiffness method for layered soils

This transformation is followed by a Fourier series expansion in the circumferentialdirection. If the displacement field is symmetric with respect to the plane θ = 0,the wave potential Φ is an even function of θ and a Fourier cosine series is used. Ifthe displacement field is antisymmetric, a Fourier sine series is used. Asymmetriccases can be decomposed into a symmetric and an antisymmetric case. Applyingthe Fourier series expansion, the partial differential equation (2.117) becomes:

(λ+ 2µ)

[

∂2

∂r2+

1

r

∂

∂r− n2

r2+

∂2

∂z2

]

Φ(r, n, z, ω)+ω2ρΦ(r, n, z, ω) = 0 (2.118)

where Φ(r, n, z, ω) denotes the n-th coefficient in the Fourier series expansion ofthe wave potential Φ(r, θ, z, ω). Equation (2.118) is a Bessel equation of ordern and can be solved by means of an n-th order Hankel transformation from theradial coordinate r to the radial wavenumber kr. This results into the followingordinary differential equation:

(λ+ 2µ)

[

−k2r +

d2

dz2

]

Φ(kr, n, z, ω) + ω2ρΦ(kr, n, z, ω) = 0 (2.119)

This equation is similar to the ordinary differential equation (2.69) governing two-dimensional P-wave propagation and a similar solution is obtained:

Φ(kr, n, z, ω) = I ′Pe−ikzpz + R′

Pe+ikzpz (2.120)

where the vertical wavenumber kzp is calculated as:

kzp = ±√

k2p − k2

r (2.121)

The sign in equation (2.121) is chosen so that Im(kzp) ≤ 0 if kzp is complex andωkzp ≥ 0 if kzp is real.

The wave potential Φ(r, θ, z, ω) in the frequency-space domain is obtained bymeans of an inverse n-th order Hankel transformation of the wavenumber kr tothe radial coordinate r followed by an inverse Fourier series expansion in thecircumferential direction:

Φ(r, θ, z, ω) =∞∑

n=0

TPn

∫ ∞

0

rJn(krr)(

I ′Pe−ikzpz + R′

Pe+ikzpz

)

dkr (2.122)

where TPn = cos(nθ) in the symmetric case and TPn = sin(nθ) in theantisymmetric case. An additional inverse Fourier transformation of the circularfrequency ω to the time t gives the wave potential Φ(r, θ, z, t) in the time-spacedomain.

Vertically polarized shear waves

The propagation of vertically polarized shear waves is governed by the hyperbolicpartial differential equation (2.62). In a similar way as for dilatational waves, the

Three-dimensional wave propagation 43

solution of this equation is obtained in the frequency-wavenumber domain as:

Ψ′(kr, n, z, ω) = I ′SVe−ikzsz + R′

SVe+ikzsz (2.123)

where the vertical wavenumber kzs is calculated as:

kzs = ±√

k2s − k2

r (2.124)

The sign in equation (2.124) is chosen so that Im(kzs) ≤ 0 if kzs is complex andωkzs ≥ 0 if kzs is real.

The wave potential Ψ′(r, θ, z, ω) in the frequency-space domain is obtained bymeans of an inverse n-th order Hankel transformation of the wavenumber kr tothe radial coordinate r followed by an inverse Fourier series expansion in thecircumferential direction:

Ψ′(r, θ, z, ω) =

∞∑

n=0

TSVn

∫ ∞

0

rJn(krr)(

I ′SVe−ikzsz + R′

SVe+ikzsz

)

dkr (2.125)

where TSVn = cos(nθ) in the symmetric case and TSVn = sin(nθ) in theantisymmetric case. An additional inverse Fourier transformation of the circularfrequency ω to the time t gives the wave potential Ψ′(r, θ, z, t) in the time-spacedomain.

Horizontally polarized shear waves

The propagation of horizontally polarized shear waves is governed by thehyperbolic partial differential equation (2.63). The solution of this equation isobtained in a similar way as for dilatational and vertically polarized shear waves.For horizontally polarized shear waves, however, the wave potential χ is an oddfunction of θ if the displacement field is symmetric with respect to the plane θ = 0and an even function of θ if the displacement field is antisymmetric with respect tothe plane θ = 0. Therefore, a Fourier sine series expansion is used in the symmetriccase and a Fourier cosine series expansion is used in the antisymmetric case. Thewave potential χ is found in the frequency-wavenumber domain as:

χ(kr, n, z, ω) = I ′SHe−ikzsz + R′

SHe+ikzsz (2.126)

where the vertical wavenumber kzs is defined in the same way as for verticallypolarized shear waves.

The wave potential χ(r, θ, z, ω) in the frequency-space domain is obtained bymeans of an inverse n-th order Hankel transformation of the wavenumber kr tothe radial coordinate r followed by an inverse Fourier series expansion in thecircumferential direction:

χ(r, θ, z, ω) =∞∑

n=0

TSHn

∫ ∞

0

rJn(krr)(

I ′SHe−ikzsz + R′

SHe+ikzsz

)

dkr (2.127)

44 The direct stiffness method for layered soils

where TSHn = sin(nθ) in the symmetric case and TSHn = cos(nθ) in theantisymmetric case. An additional inverse Fourier transformation of the circularfrequency ω to the time t gives the wave potential χ(r, θ, z, t) in the time-spacedomain.

2.4.2 Displacements

The displacements u(r, θ, z, ω) in the frequency-space domain are obtained fromthe wave potentials Φ(r, θ, z, ω), Ψ′(r, θ, z, ω), and χ(r, θ, z, ω) by means ofequations (2.58–2.60):

uruθuz

=

∂∂r −l ∂2

∂r∂z∂r∂θ

∂r∂θ −l ∂2

r∂θ∂z − ∂∂r

∂∂z l

(

∇2 − ∂2

∂z2

)

0

Φ

Ψ′

χ

(2.128)

where l is the dimensional factor previously introduced to make the potentialsΦ, Ψ′ and χ of the same dimension. Introducing equations (2.122), (2.125),and (2.127) and elaborating the partial derivatives with respect to the spatialcoordinates leads to:

u =∞∑

n=0

Tn

∫ ∞

0

kr

(

JInZ

IaI′ + JRn ZRaR′

)

dkr (2.129)

In the symmetric case, the matrix Tn is defined as:

Tn =

cos(nθ) 0 00 − sin(nθ) 00 0 cos(nθ)

(2.130)

In the antisymmetric case, the matrix Tn is defined as:

Tn =

sin(nθ) 0 00 cos(nθ) 00 0 sin(nθ)

(2.131)

The matrices ZI and ZR are defined by equation (2.90) in the same way as for two-dimensional wave propagation. The vectors aI′ and aR′ collect the wave potentialamplitudes:

aI′ =

I ′PI ′SV

I ′SH

aR′ =

R′P

R′SV

R′SH

(2.132)

Three-dimensional wave propagation 45

The matrices JIn and JR

n are defined as:

JIn =

krJ′n(krr) ilkrkzsJ

′n(krr) ±n

r Jn(krr)nr Jn(krr) ilkzs

nr Jn(krr) ±krJ ′

n(krr)−ikzpJn(krr) −lk2

rJn(krr) 0

(2.133)

JRn =

krJ′n(krr) −ilkrkzsJ ′

n(krr) ±nr Jn(krr)

nr Jn(krr) −ilkzs nr Jn(krr) ±krJ ′

n(krr)ikzpJn(krr) −lk2

rJn(krr) 0

(2.134)

where J ′n(krr) = ∂Jn(krr)/∂(krr) denotes the derivative of the Bessel function

Jn(krr) with respect to its argument krr. The minus signs in equation(2.134) correspond to the symmetric case and the plus signs correspond to theantisymmetric case. The matrices JI

n and JRn can be decomposed as:

JIn = CnD

IA (2.135)

JRn = CnD

RA (2.136)

The matrix A is defined as:

A =

i 0 00 lkr 00 0 ±i

(2.137)

where the minus sign is used in the symmetric case and the plus sign in theantisymmetric case. The matrices DI and DR are given by equation (2.98) usingkx = kr. The matrix Cn is defined as:

Cn =

J ′n(krr)

nkrr

Jn(krr) 0nkrr

Jn(krr) J ′n(krr) 0

0 0 −Jn(krr)

(2.138)

This matrix can be elaborated using the following recurrence relations:

Jn−1(krr) + Jn+1(krr) =2n

krrJn(krr) (2.139)

Jn−1(krr) − Jn+1(krr) = 2J ′n(krr) (2.140)

For the cases n = 0 and n = 1, which are used in the next chapter to calculate theGreen’s functions of a layered medium, the following expressions are obtained:

C0 =

−J1(krr) 0 00 −J1(krr) 00 0 −J0(krr)

(2.141)

C1 =

12J0(krr) − 1

2J2(krr)1krr

J1(krr) 01krr

J1(krr)12J0(krr) − 1

2J2(krr) 0

0 0 −J1(krr)

(2.142)

46 The direct stiffness method for layered soils

Using the decompositions (2.135) and (2.136), equation (2.128) is reformulated as:

u =

∞∑

n=0

Tn

∫ ∞

0

krCn

(

DIZIaI + DRZRaR)

dkr (2.143)

where the modified wave potential amplitudes aI and aR are defined as aI = AaI′

and aR = AaR′.The bracketed term in equation (2.143) is referred to as the displacement vector

u in the frequency-wavenumber domain:

u = DIZIaI + DRZRaR (2.144)

This vector u is related to the wave potential amplitudes aI and aR in the sameway as in the case of two-dimensional wave propagation.

The relation between the displacement vector u in the frequency-wavenumberdomain and the displacement vector u in the frequency-space domain follows fromequation (2.143):

u = an

∫ ∞

0

rCn

∫ 2π

0

Tnu dθdr (2.145)

u =

∞∑

n=0

Tn

∫ ∞

0

krCnu dkr (2.146)

where the coefficient an is equal to 1/(2π) if n = 0 and 1/π if n 6= 0. The dualityof the integral transformations in equations (2.145) and (2.146) can be shownby means of the recurrence relations (2.139) and (2.140) and the orthogonalityrelation:

∫ ∞

0

rJn(krr)Jn(k′rr) dr =1

krδ(kr − k′r) (2.147)

2.4.3 Tractions

The tractions tez = σzr, σzθ, σzzT on a horizontal plane with unit outwardnormal vector ez are expressed as a function of the displacements u in thefrequency-space domain using the strain-displacement relations (2.36–2.41) andthe constitutive equations (2.42–2.47):

tezr

tez

θ

tezz

=

0 0 0 0 0 2µ0 0 0 0 2µ 0λ λ λ+ 2µ 0 0 0

∂∂r 0 01r

1r∂∂θ 0

0 0 ∂∂z

12r

∂∂θ

12∂∂r − 1

2r 00 1

2∂∂z

12r

∂∂θ

12∂∂z 0 1

2∂∂r

uruθuz

Three-dimensional wave propagation 47

(2.148)

Introducing equation (2.143) and elaborating the partial derivatives with respectto the spatial coordinates leads to:

tez =∞∑

n=0

Tn

∫ ∞

0

krCn

(

GIZIaI + GRZRaR)

dkr (2.149)

where the matrices GI and GR are given by equation (2.109) using kx = kr.The bracketed term in equation (2.149) is referred to as the traction vector t

ez

in the frequency-wavenumber domain:

tez

= GIZIaI + GRZRaR (2.150)

This vector tez

is related to the wave potential amplitudes aI and aR in the sameway as in the case of two-dimensional wave propagation.

The relation between the traction vector tez

in the frequency-wavenumberdomain and the traction vector tez in the frequency-space domain follows fromequation (2.149):

tez

= an

∫ ∞

0

rCn

∫ 2π

0

Tntez dθdr (2.151)

tez =

∞∑

n=0

Tn

∫ ∞

0

krCntezdkr (2.152)

2.4.4 Strains and stresses

The strain components εrr, εθθ, and εrθ are related to the displacements u throughequations (2.36–2.41):

εrrεθθεrθ

=

∂∂r 0 01r

1r∂∂θ 0

12r

∂∂θ

12∂∂r − 1

2r 0

uruθuz

(2.153)

Equation (2.153) only involves derivatives with respect to the horizontal coordi-nates. Using equation (2.146), these derivatives are elaborated in terms of themodified displacement vector u without the use of the wave potentials aI and aR:

εrrεθθεrθ

=

∞∑

n=0

T′n

∫ ∞

0

krC′nu dkr (2.154)

In the symmetric case, the matrix T′n is defined as:

T′n =

cos(nθ) 0 00 cos(nθ) 00 0 − sin(nθ)

(2.155)

48 The direct stiffness method for layered soils

In the antisymmetric case, the matrix T′n is defined as:

T′n =

sin(nθ) 0 00 sin(nθ) 00 0 cos(nθ)

(2.156)

The matrix C′n is defined as:

C′n =

krJ′′n(krr) − n

krr2Jn(krr) + n

r J′n(krr) 0

1rJ

′n(krr) − n2

krr2Jn(krr)

nkrr2

Jn(krr) − nr J

′n(krr) 0

nr J

′n(krr) − n

krr2Jn(krr)

n2

2krr2Jn(krr) − 1

2rJ′n(krr) + kr

2 J′′n (krr) 0

(2.157)

where J ′n(krr) and J ′′

n (krr) denote the first and second derivative of the Besselfunction Jn(krr) with respect to its argument krr. For the cases n = 0 and n = 1,the matrix C′

n reduces to:

C′0 =

−krJ0(krr) + 1rJ1(krr) 0 0

− 1rJ1(krr) 0 0

0 kr

2 J2(krr) 0

(2.158)

C′1 =

−krJ1(krr) + 1rJ2(krr) − 1

rJ2(krr) 0− 1rJ2(krr)

1rJ2(krr) 0

− 1rJ2(krr) −kr

2 J1(krr) + 1rJ2(krr) 0

(2.159)

The remaining strain components εzr, εzθ, εzz and stress components σrr, σθθ, σrθare calculated in the space domain from the strain components given by equation(2.154) and the stress components given by equation (2.151) using the constitutiveequations (2.42–2.47).

2.5 Stiffness matrices and shape functions

In this section, the stiffness matrices and shape functions for a layer and a halfspaceelement are discussed. The stiffness matrices relate the displacements u and thetractions t

ezat the interfaces between elements. The shape functions allow to

interpolate the displacements u and tractions tez

at intermediate locations fromthe interface variables. The stiffness matrices and shape functions are obtainedby elimination of the wave potentials from the equations in the previous sectionsthat relate the displacements and tractions to the wave potentials. Due to the useof specific transformations from the space domain to the wavenumber domain,these equations are identical for two-dimensional and three-dimensional wavepropagation. As a result, identical stiffness matrices and shape functions areobtained in both cases.

Stiffness matrices and shape functions 49

2.5.1 Stiffness matrices

The halfspace element

The halfspace element models the propagation of waves in a semi-infinite halfspace.The amplitudes of the plane harmonic waves should be non-increasing functionswith depth. Only outgoing waves propagating in the positive z-direction aretherefore taken into account. As a result, two-dimensional wave propagation inthe element is governed by the wave potentials given by equations (2.75), (2.82),and (2.85), where the wave amplitudes RP, RSV, and RSH of the incoming wavesare equal to zero. Similarly, three-dimensional wave propagation in the element isgoverned by the wave potentials given by equations (2.122), (2.125), and (2.127),where the wave amplitudes R′

P, R′SV, and R′

SH of the incoming waves are equal tozero.

xy

z

uex1

uez1

uey1

tex1

tez1

tey1

Figure 2.3: The halfspace element.

Figure 2.3 shows the displacements ue = uex1, uey1, uez1T and tractions t

e=

tex1, tey1, tez1T at the surface of a halfspace element. The displacements ue at the

element surface are expressed in terms of the wave potentials using equation (2.97)or (2.144) where aR = 0:

ue = DIZI(z = 0)aI (2.160)

Herein, the submatrix ZI(z = 0) is an identity matrix.Similarly, the tractions t

eat the element surface are expressed in terms of the

wave potentials using equation (2.107) or (2.150) where aR = 0:

te = −GIZI(z = 0)aI (2.161)

The minus sign accounts for the negative direction of the unit outward normalvector at the surface of the element.

50 The direct stiffness method for layered soils

The element tractions te

are related to the element displacements ue after

elimination of the outgoing wave amplitudes aI from equations (2.160) and (2.161):

Keue = t

e(2.162)

where Ke is the complex symmetrical element stiffness matrix of the halfspaceelement:

Ke = −GIDI−1 (2.163)

The straightforward numerical evaluation of equation (2.163) is not advisable,as some of the submatrices are singular for zero frequency (static case) and/orhorizontal (one-dimensional case) and vertical wavenumber. In view of numericalaccuracy and computational efficiency, analytical expressions are therefore derivedfor the elements of the halfspace stiffness matrix, as well as limiting expressions forω, kx, kzp, and kzs tending to zero. These expressions are used in the MATLABtoolbox EDT 2.0, but not included in this text as they are very lengthy and provideno additional insight in the direct stiffness method. Equivalent expressions can befound in reference [113].

Due to the uncoupling of the in-plane motion and the out-of-plane motion, thecoupling terms between the x or z-direction and the y-direction vanish in thestiffness matrix Ke. The relation (2.162) between the element displacements u

e

and tractions te

can therefore be decomposed as:

KePSVu

ePSV = t

ePSV (2.164)

KeSHu

eSH = t

eSH (2.165)

where uePSV = uex1, uez1T and t

ePSV = tex1, tez1T are related to P-SV-wave

propagation and ueSH = uey1 and t

eSH = tey1 are related to SH-wave propagation.

The 2× 2 matrix KePSV and the 1× 1 matrix Ke

SH are submatrices of the elementstiffness matrix Ke. The use of the uncoupled element stiffness matrices Ke

PSV andKe

SH in the MATLAB toolbox EDT 2.0 leads to two smaller systems of equationsthat can be solved more efficiently.

The layer element

The layer element models the propagation of waves in a layer of finite thicknessL, where both incoming and outgoing waves propagate. The wave propagation inthe element is governed by the wave potentials given by equations (2.75), (2.82),and (2.85) in the two-dimensional case and equations (2.122), (2.125), and (2.127)in the three-dimensional case.

Figure 2.4 shows the displacements ue = uex1, uey1, uez1, uex2, uey2, uez2T and

tractions te

= tex1, tey1, tez1, tex2, tey2, tez2T at the boundaries of a layer element.

Stiffness matrices and shape functions 51

xy

z

uex1

uez1

tex2

tez2

uey1

tey2

tex1

tez1

uex2

uez2

tey1

uey2

L

Figure 2.4: The layer element.

The displacements ue at the element boundaries are expressed in terms of the

wave potentials using equation (2.97) or (2.144):

ue =

[

DIZI(z = 0) DRZR(z = 0)

DIZI(z = L) DRZR(z = L)

]

aI

aR

(2.166)

where the submatrices ZI(z = 0) and ZR(z = 0) are identity matrices.Similarly, the tractions t

eat the element boundaries are expressed in terms of

the wave potentials using equation (2.107) or (2.150):

te =

[

−GIZI(z = 0) −GRZR(z = 0)

GIZI(z = L) GRZR(z = L)

]

aI

aR

(2.167)

The minus sign accounts for the negative direction of the unit outward normalvector at the upper boundary of the element. Elimination of the wave amplitudesaI and aR from equations (2.166) and (2.167) allows to relate the elementdisplacements u

e to the element tractions te

as:

Keue = t

e(2.168)

or, in blockwise notation:

[

Ke11 Ke

12

Ke21 Ke

22

]

ue1

ue2

=

te1

te2

(2.169)

As Ke is symmetric, Ke12 = KeT

21 . Moreover, a symmetry argument can be usedto demonstrate that the elements of the submatrix Ke

22 follow immediately from

52 The direct stiffness method for layered soils

the elements of Ke11. As a result, only the elements of the submatrices Ke

11 andKe

21 have to be determined explicitly.It is not advisable to calculate the layer stiffness matrix as the product of the

coefficient matrix in equation (2.167) and the inverse of the coefficient matrix inequation (2.166). Due to the presence of the submatrices ZI(z = L) and ZR(z =L), the latter is severely ill-conditioned for large values of kxL, which would resultin very poor computational results [111]. Instead, matrix algebra can be used toobtain the following expressions for the submatrices Ke

11 and Ke21 in terms of the

submatrices introduced earlier:

Ke11 = −

[

GI − GRZILD

R−1DIZIL

] [

DI − DRZILD

R−1DIZIL

]−1

(2.170)

Ke21 =

[

GI − GRDR−1DI]

ZIL

[

DI − DRZILD

R−1DIZIL

]−1

(2.171)

where it is understood that the submatrix ZIL is evaluated at z = L.

Expressions (2.170) and (2.171) are stable for limiting large values of thehorizontal wavenumber kx and the layer thickness L. The submatrix Ke

11 tendsto −GIDI−1, the expression of the stiffness matrix of a halfspace element.Furthermore, the submatrix Ke

21 tends to zero and the interfaces becomeuncoupled.

In view of numerical accuracy and computational efficiency, analytical expres-sions are derived for the elements of the layer stiffness matrix, as well as limitingexpressions for ω, kx, kzp, and kzs tending to zero. While these expressions areused in the MATLAB toolbox EDT 2.0, they are omitted here as they are verylengthy and provide no additional insight. Equivalent expressions can be found inreference [113], but these expressions are not stable for large values of kxL unlessspecial precautions are taken [111].

In a similar way as for the halfspace element, the relation (2.168) between theelement displacements u

e and tractions te

can be decomposed as:

KePSVu

ePSV = t

ePSV (2.172)

KeSHu

eSH = t

eSH (2.173)

where uePSV = uex1, uez1, uex2, uez2T and t

ePSV = tex1, tez1, tex2, tez2T are related to

P-SV-wave propagation and ueSH = uey1, uey2T and t

eSH = tey1, tey2T are related

to SH-wave propagation. The 4 × 4 matrix KePSV and the 2 × 2 matrix Ke

SH aresubmatrices of the element stiffness matrix Ke.

Assembly of equations

The propagation of waves in a horizontally layered halfspace can now be modelledwith N − 1 layer elements on a halfspace element (figure 2.5). For each interface

Stiffness matrices and shape functions 53

xy

z

12

i− 1i

i+ 1N − 2N − 1N

[1][2]

[i][i+ 1]

[N − 1][N ]

Figure 2.5: A horizontally layered halfspace consisting of N − 1 layers on ahalfspace.

i (with 1 ≤ i ≤ N) between elements i− 1 and i, the continuity of displacementsis expressed in the frequency-wavenumber domain as:

ui−12 = u

i1 (2.174)

The vectors ui−12 and u

i1 collect the displacements at the lower boundary of element

i− 1 and at the upper boundary of element i, respectively. In the following of thissubsection, both vectors are commonly denoted by u

i, so that equation (2.174) isimplicitly accounted for.

For each interface i, the stress equilibrium (figure 2.6) is expressed in thefrequency-wavenumber domain as:

pi= t

i−12 + t

i1 (2.175)

The vectors ti−12 and t

i1 denote the tractions on the lower boundary of element

i − 1 and on the upper boundary of element i, respectively. The load vector pi

collects the external tractions applied at interface i.Equation (2.175) is rewritten in terms of the interface displacements u

i usingequations (2.162) and (2.169):

Ki−121 u

i−1 +(

Ki−122 + Ki

11

)

ui + Ki

12ui+1 = p

i(2.176)

where Ki11 = Ki and Ki

12 = 0 if element i is a halfspace element. Equation (2.176)is equivalent to:

Ku = p (2.177)

where the displacement vector u collects the interface displacements ui and the

load vector p collects the external tractions pi

on the interfaces. The stiffness

54 The direct stiffness method for layered soils

ti−1x2

ti−1x2

ti−1y2

ti−1y2

ti−1z2

ti−1z2

tix1

tix1

tiy1

tiy1

tiz1

tiz1

pixpiy

piz

i− 1

i

[i]

Figure 2.6: Equilibrium of interface i between elements i− 1 and i.

matrix K in equation (2.177) is assembled from the element stiffness matrices inan analogous way as in a finite element formulation:

K =

K111 K1

12

K121 K1

22 + K211 K2

12

K221 K2

22 + K311 K3

12

K321 K3

22 + K411 . . .

.... . .

(2.178)

From a computational point of view, it is advantageous to exploit the uncouplingof P-SV and SH-waves and to assemble separate stiffness matrices KPSV and KSH,using equations (2.164), (2.165), (2.172), and (2.173). The equilibrium is thenexpressed as:

KPSVuPSV = pPSV (2.179)

KSHuSH = pSH (2.180)

where uPSV and pPSV collect the displacements and tractions in x and z-directionand uSH and pSH collect the displacements and tractions in y-direction.

Stiffness matrices and shape functions 55

2.5.2 Shape functions for the displacements

The halfspace element

The displacements u(z) at a depth z in a halfspace element are expressed in termsof the wave potentials using equation (2.97) or (2.144) where aR = 0:

u(z) = DIZI(z)aI (2.181)

The displacements ue at the surface of the halfspace element are given by equation

(2.160). The displacements u(z) are related to the surface displacements ue after

elimination of the wave amplitudes aI from equations (2.160) and (2.181):

u(z) = Ne(z)ue (2.182)

where the matrix Ne(z) contains the shape functions for the displacements in thehalfspace element:

Ne(z) = DIZI(z)DI−1 (2.183)

In a similar way as for the stiffness matrices, analytical expressions are derived forthe elements of the matrix Ne(z), as well as limiting expressions for ω, kx, kzp,and kzs tending to zero.

Due to the uncoupling of P-SV-waves and the SH-waves, the coupling termsbetween the x or z-direction and the y-direction vanish in the matrix Ne(z) andequation (2.182) can be reformulated as:

uPSV(z) = NePSV(z)uePSV (2.184)

uSH(z) = NeSH(z)ueSH (2.185)

where uPSV(z) = ux(z), uz(z)T and uSH(z) = uy(z).

The layer element

The displacements u(z) at a depth z in a layer element are expressed in terms ofthe wave potentials using equation (2.97) or (2.144):

u(z) =[

DIZI(z) DRZR(z)]

aI

aR

(2.186)

The displacements ue at the boundaries of the layer element are given by equation

(2.166). Elimination of the wave potential amplitudes aI and aR from equations(2.166) and (2.186) allows to relate the displacements u(z) to the interfacedisplacements u

e:

u(z) = Ne(z)ue (2.187)

56 The direct stiffness method for layered soils

or, in blockwise notation:

u(z) =[

Ne1(z) Ne

2(z)]

ue1

ue2

(2.188)

Matrix algebra is used to obtain the following numerically stable expressions forthe submatrices Ne

1(z) and Ne2(z) in terms of the submatrices introduced earlier:

Ne1(z) =

[

DIZIz − DRZI

L−zDR−1DIZI

L

] [

DI − DRZILD

R−1DIZIL

]−1

(2.189)

Ne2(z) =

[

DRZIL−z − DIZI

zDI−1DRZI

L

] [

DR − DIZILD

I−1DRZIL

]−1

(2.190)

where the submatrices ZIz and ZI

L−z are evaluated at z and L − z, respectively.Expressions (2.189) and (2.190) are stable for limiting large values of the horizontalwavenumber kx and the layer thickness L. Analytical expressions are derived forthe elements of the matrices Ne

1(z) and Ne2(z), as well as limiting expressions for

ω, kx, kzp, and kzs tending to zero.In a similar way as for the halfspace element, equation (2.187) can be decomposed

as:

uPSV(z) = NePSV(z)uePSV (2.191)

uSH(z) = NeSH(z)ueSH (2.192)

2.5.3 Shape functions for the tractions

The halfspace element

The tractions tez

on a horizontal plane with unit outward normal vector ez at adepth z in a halfspace element are expressed in terms of the wave potentials usingequation (2.107) or (2.150) where aR = 0:

tez

(z) = GIZI(z)aI (2.193)

The displacements ue at the surface of the halfspace element are given by equation

(2.160). The tractions tez

(z) are related to the surface displacements ue after

elimination of the wave amplitudes aI from equations (2.160) and (2.193):

tez

(z) = Be(z)ue (2.194)

where the matrix Be(z) contains the shape functions for the tractions in thehalfspace element:

Be(z) = GIZI(z)DI−1 (2.195)

Analytical expressions are derived for the elements of the matrix Be(z), as well aslimiting expressions for ω, kx, kzp, and kzs tending to zero.

Stiffness matrices and shape functions 57

Due to the uncoupling of P-SV-waves and the SH-waves, the coupling termsbetween the x or z-direction and the y-direction vanish in the matrix Be(z) andequation (2.194) can be reformulated as:

tez

PSV(z) = BePSV(z)uePSV (2.196)

tez

SH(z) = BeSH(z)ueSH (2.197)

where tez

PSV(z) = tezx (z), tez

z (z)T and tez

SH(z) = tezy (z).

The layer element

The tractions tez

(z) on a horizontal plane with unit outward normal vector ez ata depth z in a layer element are expressed in terms of the wave potentials usingequation (2.97) or (2.144):

tez

(z) =[

GIZI(z) GRZR(z)]

aI

aR

(2.198)

The displacements ue at the boundaries of the layer element are given by

equation (2.166). Elimination of the wave potential amplitudes aI and aR fromequations (2.166) and (2.198) allows to relate the tractions t

ez(z) to the interface

displacements ue:

tez

(z) = Be(z)ue (2.199)

or, in blockwise notation:

tez

(z) =[

Be1(z) Be

2(z)]

ue1

ue2

(2.200)

Matrix algebra is used to obtain the following numerically stable expressions:

Be1(z) =

[

GIZIz − GRZI

L−zDR−1DIZI

L

] [

DI − DRZILD

R−1DIZIL

]−1

(2.201)

Be2(z) =

[

GRZIL−z − GIZI

zDI−1DRZI

L

] [

DR − DIZILD

I−1DRZIL

]−1

(2.202)

Analytical expressions are derived for the elements of the matrices Be1(z) and

Be2(z), as well as limiting expressions for ω, kx, kzp, and kzs tending to zero.In a similar way as for the halfspace element, equation (2.199) can be decomposed

as:

tez

PSV(z) = BePSV(z)uePSV (2.203)

tez

SH(z) = BeSH(z)ueSH (2.204)

58 The direct stiffness method for layered soils

2.6 Conclusion

In this chapter, the direct stiffness method for wave propagation in horizontallylayered elastic media is addressed.

First, the elastodynamic equations are introduced. Using a Helmholtzdecomposition of the displacement vector, uncoupled partial differential equationsare formulated for the wave potentials that govern dilatational waves andhorizontally and vertically polarized shear waves.

Next, two-dimensional wave propagation in a layered medium is addressed.Using a double forward Fourier transformation from the time-space domain tothe frequency-wavenumber domain, ordinary differential equations for the wavepotentials are obtained where the vertical coordinate is the only independentvariable. These equations yield solutions for the wave potentials in terms ofunknown wave potential amplitudes. In order to obtain these amplitudes, theyare related to the displacements and tractions on a horizontal plane.

A similar approach is followed for the case of three-dimensional wave propaga-tion. Here, the partial differential equations are solved by means of a forwardFourier transformation from the time domain to the frequency domain, followedby a Fourier series expansion in the circumferential direction and a Hankeltransformation in the radial direction.

Elimination of the unknown wave potential amplitudes from the relationsbetween wave potentials, displacements, and tractions leads to the frequency-wavenumber domain stiffness matrices and shape functions of the layer andhalfspace elements. Due to the use of specific transformations from the spacedomain to the wavenumber domain, identical results are obtained for the two-dimensional and the three-dimensional case.

The direct stiffness method is used in the next chapter to compute the two-dimensional and three-dimensional Green’s functions of a layered soil. It is alsoused in the frame of the SASW method addressed in chapter 5 to calculate thetheoretical dispersion curve of a layered soil.

Chapter 3

Green’s functions of layered

soils

3.1 Introduction

This chapter addresses the computation of the Green’s displacement and stressfunctions of a layered soil. The Green’s functions represent the dynamic responseof the soil to a unit load. They are used in the boundary integral formulation,which is discussed in the next chapter.

The Green’s functions are calculated with the direct stiffness method in thefrequency-wavenumber domain and subsequently transformed to the frequency-space domain. Different transformations are used to obtain the two-dimensionaland three-dimensional Green’s functions. In the two-dimensional case, inverseFourier transformations are used, while in the three-dimensional case, inverseHankel transformations are used.

The transformation of the Green’s functions is complicated by the occurrence ofpoles in the wavenumber domain that correspond to the surface wave modes of themedium [10]. The introduction of complex Lame coefficients in the equilibriumequations shifts the poles from the real wavenumber axis into the complexplane and enables the use of numerical integration schemes. The transformationis further complicated by the presence of a singularity in the space domainGreen’s functions as the source-receiver distance tends to zero. The wavenumberdomain Green’s functions are therefore decomposed into a singular part, which istransformed analytically, and a regular part, which is transformed numerically.

While the inverse Fourier transformation can be evaluated by means of a FastFourier Transformation (FFT) algorithm, no comparable algorithm exists for theinverse Hankel transformation. If the inverse Hankel transformation is evaluatedwith a Newton-Cotes formula, such as the trapezium rule, an appropriate choiceof the quadrature step is difficult due to the rapid oscillatory nature of the

59

60 Green’s functions of layered soils

transformation kernel (a Bessel function) for large horizontal distances betweensource and receiver. In the literature, several alternative algorithms have thereforebeen presented.

Frazer and Gettrust [72] propose a generalized Filon [69] integration scheme.The integration domain [0,∞) is truncated at a finite value and subdivided intosmall intervals. In each interval, the signal is approximated by a polynomial andthe integral is analytically evaluated.

Branders [26] subdivides the integration domain in an interval [0, k0], wherea classical quadrature scheme is used, and an interval [k0,∞), where thetransformation kernel is replaced with an asymptotic expression in terms oftrigonometric functions. The integration over the interval [k0,∞) is performedby means of a Fourier sine and a Fourier cosine transformation.

Apsel and Luco [10] use a similar subdivision of the integration domain. In theinterval [0, k0], the integrand (including the transformation kernel) is approximatedby a piecewise polynomial and the integral is analytically evaluated. In the interval[k0,∞), the transformation kernel is replaced with an asymptotic expression andthe integral is evaluated by means of Filon quadrature. The advantage over themethod proposed by Branders [26] is the possibility to adapt the sampling schemein regions where the signal is particularly sharp or smooth.

Degrande et al. [48] combine the generalized Filon integration method proposedby Frazer and Gettrust [72] with the adaptive sampling scheme used by Apsel andLuco [10] to calculate the Green’s displacement functions of layered soils.

Alternative methods are based on a logarithmic change of variables to express theHankel transformation as a convolution. Anderson [8, 9] applies a digital filter tocompute this convolution. Talman [212] uses the convolution theorem and makesefficient use of an FFT algorithm. He considers both the Hankel transformationand the Fourier transformation in logarithmic variables.

Algorithms based on a logarithmic sampling scheme are particularly efficient forproblems involving considerably different length scales. This is the case in dynamicsoil-structure interaction problems, where both the interaction between adjacentpoints on the soil-structure interface and the radiation of waves to the far fieldare of interest. In the present work, Talman’s method is therefore used for theinverse Fourier and Hankel transformations in the calculation of the space domainGreen’s functions. The original algorithm is improved through the use of a windowand a filter to mitigate artifacts caused by the Gibbs phenomenon. The algorithmis included in the MATLAB toolbox EDT 2.0, where it is used to compute thetwo-dimensional and three-dimensional Green’s displacement and stress functionsof layered soils.

This chapter is structured as follows. In section 3.2, expressions for the two-dimensional Green’s functions are derived. These expressions are relatively simpledue to the straightforward relation between the space domain and the wavenumberdomain in the two-dimensional case. However, the two-dimensional Green’sfunctions are treated in detail as they serve as basis functions for the wavenumberdomain representation of the wave fields due to any load distribution in two-

Two-dimensional Green’s functions 61

dimensional or three-dimensional space. In section 3.3, expressions for the three-dimensional Green’s functions are derived in terms of the two-dimensional Green’sfunctions considered in section 3.2. Subsequently, the logarithmic Fourier andHankel transformation algorithms are detailed in sections 3.4 and 3.5. Finally, anumerical example is presented in section 3.6.

3.2 Two-dimensional Green’s functions

The two-dimensional Green’s displacement tensor of a layered halfspace is denotedby uG

ij(x′, z′, x, z, ω). This tensor represents the displacements uj(x, z, ω) at the

position (x, z) in the direction ej due to a harmonic line load pj(x, z, ω) =δijδ(x−x′)δ(z−z′) applied at the position (x′, z′) in the direction ei. As the systemgeometry and the material properties are invariant in the horizontal direction, itcan be assumed without loss of generality that the load is applied at the position(0, z′). The notation of the Green’s displacement tensor therefore reduces touGij(z

′, x, z, ω). Similarly, the two-dimensional Green’s stress and strain tensors

of a layered halfspace are denoted by σGijk(z

′, x, z, ω) and εGijk(z′, x, z, ω). These

tensors represent the stresses σjk(x, z, ω) and strains εjk(x, z, ω) at the position(x, z) due to a harmonic line load pj(x, z, ω) = δijδ(x)δ(z − z′) applied at theposition (0, z′) in the direction ei.

3.2.1 Excitation in the x-direction

The Green’s functions uGxj(z

′, x, z, ω), σGxjk(z

′, x, z, ω), and εGxjk(z′, x, z, ω) repre-

sent the displacements, stresses, and strains due to a harmonic line load at theposition (0, z′) in the x-direction. In the frequency-space domain, the load vectorp(x, z, ω) is given by:

p(x, z, ω) =

δ(x)δ(z − z′)00

(3.1)

This vector is transformed to the wavenumber domain according to equation(2.105):

p(kx, z, ω) =

δ(z − z′)00

(3.2)

The resulting displacements u and tractions tez

are calculated in the frequency-wavenumber domain by means of the direct stiffness formulation. First, the layeredhalfspace is modelled with layer elements and a halfspace element. If necessary,an extra interface is provided at the depth z′ where the load is applied. Next, theinterface displacements are obtained using equations (2.179) and (2.180). Finally,

62 Green’s functions of layered soils

the displacements and tractions at locations within elements are calculated fromthe interface displacements by means of the shape functions discussed in sections2.5.2. and 2.5.3. By definition, these displacements and tractions are the Green’sfunctions in the frequency-wavenumber domain:

u(kx, z, ω) =

uGxx(z

′, kx, z, ω)0

uGxz(z

′, kx, z, ω)

(3.3)

tez

(kx, z, ω) =

tGezxx (z′, kx, z, ω)

0tGezxz (z′, kx, z, ω)

(3.4)

The y-components of the vectors u and tez

vanish due to the uncoupling of P-SV-waves and SH-waves.

The Green’s functions uGxj and t

Gez

xj in equations (3.3) and (3.4) are the modifiedGreen’s functions, in the sense that the vertical component is multiplied with theimaginary unit i. These equations are reformulated in terms of the unmodifiedGreen’s functions uG

xj and tGez

xj according to equations (2.93) and (2.104):

u(kx, z, ω) =

uGxx(z

′, kx, z, ω)0

iuGxz(z

′, kx, z, ω)

(3.5)

tez

(kx, z, ω) =

tGezxx (z′, kx, z, ω)

0itGezxz (z′, kx, z, ω)

(3.6)

The wavenumber domain displacements in equation (3.5) are transformed backto the space domain according to equation (2.96). This results in the followingexpressions for the two-dimensional Green’s displacement functions:

uGxx = F−1

[

uGxx;x

]

(3.7)

uGxy = 0 (3.8)

uGxz = F−1

[

uGxz;x

]

(3.9)

The wavenumber domain tractions in equation (3.6) are transformed to the spacedomain according to equation (2.106). This leads to the space domain tractionstezx , tez

y , and tezz , or, equivalently, the Green’s stress functions σG

xzx, σGxzy, and σG

xzz :

σGxzx = F−1

[

tGezxx ;x

]

(3.10)

σGxzy = 0 (3.11)

σGxzz = F−1

[

tGezxz ;x

]

(3.12)

Two-dimensional Green’s functions 63

The Green’s strain functions εGxxx, εGxyy, and εGxxy are calculated from thewavenumber domain displacements in equation (3.5) by means of equation (2.111):

εGxxx = F−1[

−ikxuGxx;x

]

(3.13)

εGxyy = 0 (3.14)

εGxxy = 0 (3.15)

The remaining Green’s stress functions σGxxx, σ

Gxyy, and σG

xxy and Green’s strain

functions εGxzx, εGxzy, and εGxzz are obtained in the space domain by means of the

constitutive equations (2.6) from the stresses σGxzx, σ

Gxzy, and σG

xzz and the strains

εGxxx, εGxyy, and εGxxy.

3.2.2 Excitation in the y-direction

The Green’s functions uGyj(z

′, x, z, ω), σGyjk(z

′, x, z, ω), and εGyjk(z′, x, z, ω) repre-

sent the displacements, stresses, and strains due to a harmonic line load at theposition (0, z′) in the y-direction. In the frequency-space domain, the load vectorp(x, z, ω) is given by:

p(x, z, ω) =

0δ(x)δ(z − z′)

0

(3.16)

This vector is transformed to the wavenumber domain according to equation(2.105):

p(kx, z, ω) =

0δ(z − z′)

0

(3.17)

By definition, the resulting displacements and tractions are the Green’s functionsin the frequency-wavenumber domain:

u(kx, z, ω) =

0uGyy(z

′, kx, z, ω)0

=

0uGyy(z

′, kx, z, ω)0

(3.18)

tez

(kx, z, ω) =

0tGezyy (z′, kx, z, ω)

0

=

0tGezyy (z′, kx, z, ω)

0

(3.19)

The x and z-components of the vectors u and tez

vanish due to the uncoupling ofP-SV-waves and SH-waves.

64 Green’s functions of layered soils

The wavenumber domain displacements in equation (3.18) are transformed backto the space domain according to equation (2.96). This results in the followingexpressions for the two-dimensional Green’s displacement functions:

uGyx = 0 (3.20)

uGyy = F−1

[

uGyy;x

]

(3.21)

uGyz = 0 (3.22)

Similarly, the wavenumber domain tractions in equation (3.19) are transformedaccording to equation (2.106). This leads to the space domain tractions tez

x , tezy ,

and tezz , or, equivalently, the Green’s stress functions σG

yzx, σGyzy, and σG

yzz :

σGyzx = 0 (3.23)

σGyzy = F−1

[

tGezyy ;x

]

(3.24)

σGyzz = 0 (3.25)

The Green’s strain functions εGyxx, εGyyy, and εGyxy are calculated from thewavenumber domain displacements in equation (3.18) by means of equation(2.111):

εGyxx = 0 (3.26)

εGyyy = 0 (3.27)

εGyxy =1

2F−1

[

−ikxuGyy;x

]

(3.28)

The remaining Green’s stress functions σGyxx, σ

Gyyy, and σG

yxy and Green’s strain

functions εGyzx, εGyzy, and εGyzz are obtained in the space domain by means of the

constitutive equations (2.6) from the stresses σGyzx, σ

Gyzy, and σG

yzz and the strains

εGyxx, εGyyy, and εGyxy.

3.2.3 Excitation in the z-direction

The Green’s functions uGzj(z

′, x, z, ω), σGzjk(z

′, x, z, ω), and εGzjk(z′, x, z, ω) repre-

sent the displacements, stresses, and strains due to a harmonic line load at theposition (0, z′) in the z-direction. In the frequency-space domain, the load vectorp(x, z, ω) is given by:

p(x, z, ω) =

00

δ(x)δ(z − z′)

(3.29)

Two-dimensional Green’s functions 65

This vector is transformed to the wavenumber domain according to equation(2.105):

p(kx, z, ω) =

00

iδ(z − z′)

(3.30)

The vector p is the modified load vector, hence the imaginary unit i in the verticalcomponent.

By definition, the resulting displacements and tractions are the Green’s functionsin the frequency-wavenumber domain:

u(kx, z, ω) =

uGzx(z

′, kx, z, ω)0

uGzz(z

′, kx, z, ω)

=

uGzx(z

′, kx, z, ω)0

iuGzz(z

′, kx, z, ω)

(3.31)

tez

(kx, z, ω) =

tGezzx (z′, kx, z, ω)

0tGezzz (z′, kx, z, ω)

=

tGezzx (z′, kx, z, ω)

0itGezzz (z′, kx, z, ω)

(3.32)

The wavenumber domain displacements in equation (3.31) are transformed backto the space domain according to equation (2.96). This results in the followingexpressions for the two-dimensional Green’s displacement functions:

uGzx = F−1

[

uGzx;x

]

(3.33)

uGzy = 0 (3.34)

uGzz = F−1

[

uGzz;x

]

(3.35)

Similarly, the wavenumber domain tractions in equation (3.32) are transformedaccording to equation (2.106). This leads to the space domain tractions tez

x , tezy ,

and tezz , or, equivalently, the Green’s stress functions σG

zzx, σGzzy , and σG

zzz :

σGzzx = F−1

[

tGezzx ;x

]

(3.36)

σGzzy = 0 (3.37)

σGzzz = F−1

[

tGezzz ;x

]

(3.38)

The Green’s strain functions εGzxx, εGzyy, and εGzxy are calculated from the wavenum-

ber domain displacements in equation (3.31) by means of equation (2.111):

εGzxx = F−1[

−ikxuGzx;x

]

(3.39)

εGzyy = 0 (3.40)

εGzxy = 0 (3.41)

66 Green’s functions of layered soils

The remaining Green’s stress functions σGzxx, σ

Gzyy , and σG

zxy and Green’s strain

functions εGzzx, εGzzy, and εGzzz are obtained in the space domain by means of the

constitutive equations (2.6) from the stresses σGzzx, σ

Gzzy , and σG

zzz and the strains

εGzxx, εGzyy, and εGzxy.

3.3 Three-dimensional Green’s functions

The three-dimensional Green’s displacement tensor of a layered halfspace isdenoted by uG

ij(x′, y′, z′, x, y, z, ω). This tensor represents the displacements

uj(x, y, z, ω) at the position (x, y, z) in the direction ej due to a harmonic pointload pj(x, y, z, ω) = δijδ(x−x′)δ(y− y′)δ(z− z′) applied at the position (x′, y′, z′)in the direction ei. As the system geometry and the material properties areinvariant in the horizontal direction, it can be assumed without loss of generalitythat the load is applied at the position (0, 0, z′). The notation of the Green’sdisplacement tensor therefore reduces to uG

ij(z′, x, y, z, ω). Similarly, the three-

dimensional Green’s stress and strain tensors of a layered halfspace are denotedby σG

ijk(z′, x, y, z, ω) and εGijk(z

′, x, y, z, ω). These tensors represent the stressesσjk(x, y, z, ω) and strains εjk(x, y, z, ω) at the position (x, y, z) due to a harmonicpoint load pj(x, y, z, ω) = δijδ(x)δ(y)δ(z − z′) applied at the position (0, 0, z′) inthe direction ei.

3.3.1 Excitation in the x-direction

The Green’s functions uGxj(z

′, x, y, z, ω), σGxjk(z

′, x, y, z, ω), and εGxjk(z′, x, y, z, ω)

represent the displacements, stresses, and strains due to a harmonic point load atthe position (0, 0, z′) in the x-direction. In the frequency-space domain, the loadvector p(x, y, z, ω) is given by:

p(x, y, z, ω) =

δ(x)δ(y)δ(z − z′)00

(3.42)

or, in cylindrical coordinates:

p(r, θ, z, ω) =

δ(r)δ(θ)δ(z − z′)/r00

(3.43)

This vector is transformed to the wavenumber domain according to equation(2.151), using the definition of Tn for symmetric cases:

p(kr, n, z, ω) =

12π δ1nδ(z − z′)12π δ1nδ(z − z′)

0

(3.44)

Three-dimensional Green’s functions 67

The Kronecker delta δ1n equals 1 if n = 1 and 0 otherwise. Only the termcorresponding to a circumferential wavenumber n = 1 differs from zero. The loadvector p in equation (3.44) is a linear combination of the load vectors considered insubsections 3.2.1 and 3.2.2, both multiplied with a factor 1

2π δ1n. As a consequence,

the resulting displacements u and tractions tez

are also linear combinations of thedisplacements and tractions calculated in subsections 3.2.1 and 3.2.2, and can beexpressed in terms of the two-dimensional Green’s functions in the wavenumberdomain:

u(kr, n, z, ω) =

12π δ1nu

Gxx(z

′, kr, z, ω)12π δ1nu

Gyy(z

′, kr, z, ω)12π δ1niu

Gxz(z

′, kr, z, ω)

(3.45)

tez

(kr, n, z, ω) =

12π δ1nt

Gezxx (z′, kr, z, ω)

12π δ1nt

Gezyy (z′, kr, z, ω)

12π δ1nit

Gezxz (z′, kr, z, ω)

(3.46)

The wavenumber domain displacements in equation (3.45) are transformed backto the space domain according to equation (2.146), using the definition of Tn forsymmetric cases. This results in the following expressions for the three-dimensionalGreen’s displacement functions:

uGxr =

1

4π

(

H−10

[

uGxx + uG

yy; r]

−H−12

[

uGxx − uG

yy; r]

)

cos θ (3.47)

uGxθ = − 1

4π

(

H−10

[

uGxx + uG

yy; r]

+ H−12

[

uGxx − uG

yy; r]

)

sin θ (3.48)

uGxz = − 1

2πH−1

1

[

iuGxz; r

]

cos θ (3.49)

where H−1n denotes the inverse Hankel transformation of order n. Similarly, the

wavenumber domain tractions in equation (3.46) are transformed according toequation (2.152). This leads to the space domain tractions tez

r , tez

θ , and tezz , or,

equivalently, the Green’s stress functions σGxzr, σ

Gxzθ, and σG

xzz:

σGxzr =

1

4π

(

H−10

[

tGezxx + tGez

yy ; r]

−H−12

[

tGezxx − tGez

yy ; r]

)

cos θ (3.50)

σGxzθ = − 1

4π

(

H−10

[

tGezxx + tGez

yy ; r]

+ H−12

[

tGezxx − tGez

yy ; r]

)

sin θ (3.51)

σGxzz = − 1

2πH−1

1

[

itGezxz ; r

]

cos θ (3.52)

The Green’s strain functions εGxrr, εGxθθ, and εGxrθ are calculated from thewavenumber domain displacements in equation (3.45) by means of equation

68 Green’s functions of layered soils

(2.154), using the definition of T′n for symmetric cases:

εGxrr = − 1

2π

(

H−11

[

kruGxx; r

]

− 1

rH−1

2

[

uGxx − uG

yy; r]

)

cos θ (3.53)

εGxθθ = − 1

2π

1

rH−1

2

[

uGxx − uG

yy; r]

cos θ (3.54)

εGxrθ =1

2π

(1

2H−1

1

[

kruGyy; r

]

+1

rH−1

2

[

uGxx − uG

yy; r]

)

sin θ (3.55)

The remaining Green’s stress functions σGxrr, σ

Gxθθ, and σG

xrθ and Green’s strainfunctions εGxzr, ε

Gxzθ, and εGxzz are obtained in the space domain by means of the

constitutive equations (2.42–2.47) from the stresses σGxzr, σ

Gxzθ, and σG

xzz and thestrains εGxxx, ε

Gxθθ, and εGxrθ.

Finally, the three-dimensional Green’s functions in Cartesian coordinates areobtained as:

uGxx

uGxy

uGxz

= RT

uGxr

uGxθ

uGxz

(3.56)

σGxxx σG

xxy σGxxz

σGxyx σG

xyy σGxyz

σGxzx σG

xzy σGxzz

= RT

σGxrr σG

xrθ σGxrz

σGxθr σG

xθθ σGxθz

σGxzr σG

xzθ σGxzz

R (3.57)

εGxxx εGxxy εGxxzεGxyx εGxyy εGxyzεGxzx εGxzy εGxzz

= RT

εGxrr εGxrθ εGxrzεGxθr εGxθθ εGxθzεGxzr εGxzθ εGxzz

R (3.58)

where the transformation matrix R is defined as:

R =

cos θ sin θ 0− sin θ cos θ 0

0 0 1

(3.59)

3.3.2 Excitation in the y-direction

The Green’s functions uGyj(z

′, x, y, z, ω), σGyjk(z

′, x, y, z, ω), and εGyjk(z′, x, y, z, ω)

represent the displacements, stresses, and strains due to a harmonic point load atthe position (0, 0, z′) in the y-direction. In the frequency-space domain, the loadvector p(x, y, z, ω) is given by:

p(x, y, z, ω) =

0δ(x)δ(y)δ(z − z′)

0

(3.60)

Three-dimensional Green’s functions 69

or, in cylindrical coordinates:

p(r, θ, z, ω) =

0δ(r)δ(θ)δ(z − z′)/r

0

(3.61)

This vector is transformed to the wavenumber domain according to equation(2.151), using the definition of Tn for antisymmetric cases:

p(kr, n, z, ω) =

12π δ1nδ(z − z′)12π δ1nδ(z − z′)

0

(3.62)

The wavenumber domain representation of the load vector is the same as for anexcitation in the x-direction. As a consequence, the resulting displacements andtractions are also identical:

u(kr, n, z, ω) =

12π δ1nu

Gxx(z

′, kr, z, ω)12π δ1nu

Gyy(z

′, kr, z, ω)12π δ1niu

Gxz(z

′, kr, z, ω)

(3.63)

tez

(kr, n, z, ω) =

12π δ1nt

Gezxx (z′, kr, z, ω)

12π δ1nt

Gezyy (z′, kr, z, ω)

12π δ1nit

Gezxz (z′, kr, z, ω)

(3.64)

The wavenumber domain displacements in equation (3.63) are transformed backto the space domain according to equation (2.146), using the definition of Tn

for antisymmetric cases. This results in the following expressions for the three-dimensional Green’s displacement functions:

uGyr =

1

4π

(

H−10

[

uGxx + uG

yy; r]

−H−12

[

uGxx − uG

yy; r]

)

sin θ (3.65)

uGyθ =

1

4π

(

H−10

[

uGxx + uG

yy; r]

+ H−12

[

uGxx − uG

yy; r]

)

cos θ (3.66)

uGyz = − 1

2πH−1

1

[

iuGxz; r

]

sin θ (3.67)

Similarly, the wavenumber domain tractions in equation (3.64) are transformedaccording to equation (2.152). This leads to the space domain tractions tez

r , tez

θ ,and tez

z , or, equivalently, the Green’s stress functions σGyzr, σ

Gyzθ, and σG

yzz:

σGyzr =

1

4π

(

H−10

[

tGezxx + tGez

yy ; r]

−H−12

[

tGezxx − tGez

yy ; r]

)

sin θ (3.68)

σGyzθ =

1

4π

(

H−10

[

tGezxx + tGez

yy ; r]

+ H−12

[

tGezxx − tGez

yy ; r]

)

cos θ (3.69)

σGyzz = − 1

2πH−1

1

[

itGezxz ; r

]

sin θ (3.70)

70 Green’s functions of layered soils

The Green’s strain functions εGyrr, εGyθθ, and εGyrθ are calculated from thewavenumber domain displacements in equation (3.63) by means of equation(2.154), using the definition of T′

n for antisymmetric cases:

εGyrr = − 1

2π

(

H−11

[

kruGxx; r

]

− 1

rH−1

2

[

uGxx − uG

yy; r]

)

sin θ (3.71)

εGyθθ = − 1

2π

1

rH−1

2

[

uGxx − uG

yy; r]

sin θ (3.72)

εGyrθ = − 1

2π

(1

2H−1

1

[

kruGyy; r

]

+1

rH−1

2

[

uGxx − uG

yy; r]

)

cos θ (3.73)

The remaining Green’s stress functions σGyrr, σ

Gyθθ, and σG

yrθ and Green’s strain

functions εGyzr, εGyzθ, and εGyzz are obtained in the space domain by means of the

constitutive equations (2.42–2.47) from the stresses σGyzr, σ

Gyzθ, and σG

yzz and the

strains εGyxx, εGyθθ, and εGyrθ.

Finally, the three-dimensional Green’s functions in Cartesian coordinates areobtained in a similar way as for an excitation in the x-direction.

3.3.3 Excitation in the z-direction

The Green’s functions uGzj(z

′, x, y, z, ω), σGzjk(z

′, x, y, z, ω), and εGzjk(z′, x, y, z, ω)

represent the displacements, stresses, and strains due to a harmonic point load atthe position (0, 0, z′) in the z-direction. In the frequency-space domain, the loadvector p(x, y, z, ω) is given by:

p(x, y, z, ω) =

00

δ(x)δ(y)δ(z − z′)

(3.74)

or, in cylindrical coordinates:

p(r, θ, z, ω) =

00

δ(r)δ(θ)δ(z − z′)/r

(3.75)

This vector is transformed to the wavenumber domain according to equation(2.151), using the definition of Tn for symmetric cases:

p(kr, n, z, ω) =

00

− 12π δ0nδ(z − z′)

(3.76)

Herein, the Kronecker delta δ0n equals 1 if n = 0 and 0 otherwise. Only the termcorresponding to a circumferential wavenumber n = 0 differs from zero. The loadvector p in equation (3.76) is equal to the load vector considered in subsection 3.2.3

Three-dimensional Green’s functions 71

multiplied with a factor 12π δ0ni. As a consequence, the resulting displacements

u and tractions tez

are equal to the displacements and tractions calculated insubsection 3.2.3 multiplied with the same factor, and can be expressed in terms ofthe two-dimensional Green’s functions in the wavenumber domain:

u(kr, n, z, ω) =

12π δ0niu

Gzx(z

′, kr, z, ω)0

− 12π δ0nu

Gzz(z

′, kr, z, ω)

(3.77)

tez

(kr, n, z, ω) =

12π δ0nit

Gezzx (z′, kr, z, ω)

0− 1

2π δ0ntGezzz (z′, kr, z, ω)

(3.78)

The wavenumber domain displacements in equation (3.77) are transformed backto the space domain according to equation (2.146), using the definition of Tn forsymmetric cases. This results in the following expressions for the three-dimensionalGreen’s displacement functions:

uGzr = − 1

2πH−1

1

[

iuGzx; r

]

(3.79)

uGzθ = 0 (3.80)

uGzz =

1

2πH−1

0

[

uGzz; r

]

(3.81)

Similarly, the wavenumber domain tractions in equation (3.78) are transformedaccording to equation (2.152). This leads to the space domain tractions tez

r , tez

θ ,and tez

z , or, equivalently, the Green’s stress functions σGzzr, σ

Gzzθ, and σG

zzz :

σGzzr = − 1

2πH−1

1

[

itGezzx ; r

]

(3.82)

σGzzθ = 0 (3.83)

σGzzz =

1

2πH−1

0

[

tGezzz ; r

]

(3.84)

The Green’s strain functions εGzrr, εGzθθ, and εGzrθ are calculated from thewavenumber domain displacements in equation (3.77) by means of equation(2.154), using the definition of T′

n for symmetric cases:

εGzrr = − 1

2π

(

H−10

[

ikruGzx; r

]

− 1

rH−1

1

[

iuGzx; r

]

)

(3.85)

εGzθθ = − 1

2π

1

rH−1

1

[

iuGzx; r

]

(3.86)

εGzrθ = 0 (3.87)

72 Green’s functions of layered soils

The remaining Green’s stress functions σGzrr, σ

Gzθθ, and σG

zrθ and Green’s strainfunctions εGzzr, ε

Gzzθ, and εGzzz are obtained in the space domain by means of the

constitutive equations (2.42–2.47) from the stresses σGzzr, σ

Gzzθ, and σG

zzz and thestrains εGzxx, ε

Gzθθ, and εGzrθ.

Finally, the three-dimensional Green’s functions in Cartesian coordinates areobtained in a similar way as for an excitation in the x-direction.

3.4 Numerical Fourier transformations

The calculation of the two-dimensional Green’s functions of a layered mediuminvolves the evaluation of an integral transformation of the following type:

f(x) =1

2π

∫ ∞

−∞

e−ikxxf(kx) dkx (3.88)

This inverse Fourier transformation is complicated by the presence of a singularityin the space domain Green’s functions as the source-receiver distance R =√

x2 + |z − z′|2 tends to zero. For the Green’s displacements, the singularity isonly logarithmic and the numerical integration scheme discussed in the followingcan proceed without special measures. For the Green’s strains and stresses, afirst order singularity occurs. Therefore, the function f(kx) is decomposed intoa singular part, which is transformed analytically, and a regular part, which istransformed numerically. The singular part h(kx) of the function f(kx) has thesame behaviour as the stress field in a homogeneous fullspace due to a static lineload [215] and is given in the wavenumber domain by:

h(kx) = (α1 + α2|kx||z − z′| + β1sign(kx) + β2kx|z − z′|) e−|kx||z−z′| (3.89)

The coefficients α1 and α2 vanish for odd functions h(kx), while the coefficients β1

and β2 vanish for even functions h(kx). For high wavenumbers |kx|, the functionf(kx) is dominated by its singular part h(kx). The coefficients α1, α2, β1, andβ2 are therefore estimated by fitting the singular function h(kx) to the function

f(kx) in the high wavenumber range. The function h(x) in the space domain issubsequently obtained as the inverse Fourier transformation of equation (3.89):

h(x) = α1|z − z′|πR2

+α2|z − z′|(|z − z′|2 − x2)

πR4−β1

ix

πR2−β2

2i|z − z′|2xπR4

(3.90)

The method developed by Talman [212] is used to evaluate the inverse Fouriertransformation of (the regular part of) the function f(kx). This method isbased on the FFT algorithm and is very efficient if the Green’s functions haveto be calculated at a large number of receiver points, as in a boundary elementformulation. Talman uses a logarithmic sampling scheme in both the wavenumberand space domain. The method is therefore advantageous for problems involvingconsiderably different length scales. This is the case for dynamic soil-structure

Numerical Fourier transformations 73

interaction problems. The soil impedance is affected by the interaction betweenadjacent points on the soil-structure interface and is calculated from the Green’sfunctions for small source-receiver distances. Conversely, the radiation of wavesto the far field is calculated from the Green’s functions for large source-receiverdistances. As a result, the Green’s functions have to be calculated for a wide rangeof source-receiver distances.

Using the equality e−ikxx = cos(kxx) − i sin(kxx), the inverse Fouriertransformation (3.88) is first reformulated in terms of the following integrals:

fc(x) =

∫ ∞

0

cos(kxx)f (kx) dkx (3.91)

fs(x) =

∫ ∞

0

sin(kxx)f(kx) dkx (3.92)

The numerical evaluation of equation (3.91) is considered in the following.Equation (3.92) can be addressed in a similar way.

A change of variables is performed using x = x0ev and kx = k0e

−w, so thatequation (3.91) becomes:

fc(x0ev) =

∫ ∞

−∞

cos(k0x0ev−w)f(k0e

−w)k0e−w dw (3.93)

= k0e− v

2

∫ ∞

−∞

cos(k0x0ev−w)e

v−w2 f(k0e

−w)e−w2 dw (3.94)

The reference wavenumber k0 and distance x0 may be chosen arbitrarily. In thepresent work, k0 = 1/x0 and x0 =

√xminxmax, where xmin and xmax are the

bounds of the spatial interval of interest. The factor e−v2 e

v2 is introduced in

equation (3.94) to obtain more tractable expressions in the following. Equation(3.94) is reformulated as:

fc(x0ew) = k0e

−w2 χ(w) (3.95)

where the function χ(w) is defined as the following convolution:

χ(w) = ϕ(w) ∗ ψ(w) (3.96)

The functions ϕ(w) and ψ(w) are given by:

ϕ(w) = f(k0e−w)e−

w2 (3.97)

ψ(w) = cos(k0x0ew)e

w2 (3.98)

The convolution theorem is used to evaluate equation (3.96). The Fouriertransformations of the kernel ψ(w) and the function ϕ(w) are calculated

74 Green’s functions of layered soils

analytically and with an FFT algorithm, respectively. The product of both Fouriertransformations is transformed back by means of an inverse FFT in order to obtainthe function χ(w). The use of an FFT algorithm requires that the w-axis issampled with a constant sampling interval ∆w. This implies that the kx-axis andthe x-axis must be logarithmically sampled, so that:

ln

(

kn+1

kn

)

= ∆w (3.99)

ln

(

xn+1

xn

)

= ∆w (3.100)

The Fourier transformation Ψ(s) of the kernel ψ(w) is given by:

Ψ(s) =1

2

(

(ik0x0)− 1

2 +is+ (−ik0x0)

− 12+is

)

Γ

(

1

2− is

)

(3.101)

Due to the introduction of the factor e−v2 e

v2 in equation (3.94), it is possible to

reformulate this equation as [1]:

Ψ(s) =

√

π

2(k0x0)

is− 12 eiIm(ln(Γ( 1

2−is)))+tan−1(tanh(π2 s)) (3.102)

Compared to equation (3.101), equation (3.102) is numerically more stable [212].

(a)0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

Normalized w−coordinate [ − ]

Mag

nitu

de [

− ]

(b)0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

Normalized s−coordinate [ − ]

Mag

nitu

de [

− ]

Figure 3.1: (a) Window applied to the function ϕ(w) and (b) filter applied to thekernel Ψ(s).

Equation (3.102) indicates that the spectral content Ψ(s) of the kernel ψ(w)has a constant modulus. Truncating Ψ(s) for the application of the inverse FFTleads to a sharp cut-off. As a consequence, the results of the FFT are severelyaffected by the Gibbs phenomenon [11] unless proper precautions are taken. Thefunction ϕ(w) is therefore multiplied with the window shown in figure 3.1a. Theshape of this window corresponds to the modulus of the spectrum of a third order

Numerical Fourier transformations 75

band-pass Butterworth filter with cut-off frequencies at 0.02 and 0.98 times theNyquist frequency. Furthermore, the spectral content Ψ(s) of the kernel ψ(w) ismultiplied with the filter shown in figure 3.1b. This filter is defined as the modulusof a fifth order low-pass Butterworth filter with a cut-off frequency of 0.9 timesthe Nyquist frequency. The exact characteristics of the window and the filter havelittle influence on the results.

−10 −5 0 5 10−20

−10

0

10

20

w

ψ(w

)

Figure 3.2: Kernel ψ(w) of the logarithmic Fourier cosine transformation.

Figure 3.2 shows the filtered kernel ψ(w). The function ψ(w) is calculated fromΨ(s) by means of an inverse FFT using 8192 samples and a sampling interval∆w = 0.009217. This inverse Fourier transformation is not a part of Talman’salgorithm, it is only performed here to create figure 3.2. Figure 3.2 shows thatthe filtered kernel ψ(w) only (significantly) differs from zero in a bounded window

[wψmin, wψmax]. For small values of w, the kernel ψ(w) decays exponentially. The

kernel ψ(w) does not exceed a threshold value ε if w < wψmin where wψmin followsfrom equation (3.98) as:

wψmin = 2 ln(ε) (3.103)

For large values of w, the kernel ψ(w) vanishes due to the application of the low-pass filter that limits the spectral content Ψ(s) of the kernel ψ(w) to the band[−π/∆w, π/∆w]. The kernel ψ(w) vanishes if its instantaneous frequency k0x0e

w

exceeds the frequency π/∆w, or if w > wψmax where wψmax is given by:

wψmax = ln

(

π

γk0x0∆w

)

(3.104)

Herein, the factor γ ≤ 1 is introduced to account for the deviation of the kernelψ(w) from a harmonic function. This deviation causes the kernel ψ(w) not tovanish immediately when its instantaneous frequency exceeds the frequency π/∆w.In the present work, a threshold value ε = 10−6 and a factor γ = 0.5 are used.

The convolution χ(w) defined by equation (3.96) can be interpreted as a weightedaverage of the function ϕ(v) where the weighting function is the kernel ψ(w−v). As

76 Green’s functions of layered soils

the kernel (almost) vanishes outside the window [wψmin, wψmax], the function χ(w) at

the point w is only affected by the function ϕ(v) in the window [w−wψmax, w+wψmin].

Similarly, the wavenumber range [kmin, kmax] where the function f(kx) must besampled can be expressed in terms of the spatial interval [xmin, xmax] of interest.Using equations (3.103) and (3.104), the following expressions are obtained:

kmin =k0x0ε

2

xmax(3.105)

kmax =π

xminγ∆w(3.106)

As the variation of the wavenumber domain Green’s functions is smooth for verylow and very high wavenumbers, the function f(kx) may be calculated in a smallerwavenumber range and extrapolated up to kmin and kmax. This approach isfollowed in the present work. Since the kernel ψ(w) is not exactly zero outside

the window [wψmin, wψmax], the bounds given by equations (3.105) and (3.106) are

sufficient only if the magnitude of the function f(kx) remains limited outside therange [kmin, kmax]. If needed, this range has to be extended, so that it includes thepeaks in the wavenumber domain Green’s functions corresponding to the surfacewave modes.

Within the wavenumber range [kmin, kmax], the function f(kx) is logarithmically

sampled, so that ln(kn+1

kn) = ∆w. The sampling interval ∆w has to be chosen

so that the peaks in the wavenumber domain Green’s functions are properlyrepresented. The following rule of thumb can be used:

∆w = ln

(

1 + βminCmin

Cmax

)

≈ βminCmin

Cmax(3.107)

where βmin, Cmin, and Cmax respectively denote the lowest hysteretic dampingratio, the lowest (shear) wave velocity, and the highest (dilatational) wave velocityin the soil model. Equation (3.107) is an adaptation of a rule of thumb proposed byClouteau [38] for the case where the wavenumber axis is sampled with a constantinterval, accounting for the fact that the Green’s functions are smooth and thesampling can be coarser in the wavenumber range above approximately ω/Cmin.

3.5 Numerical Hankel transformations

The calculation of the three-dimensional Green’s functions of a layered mediuminvolves the evaluation of an integral transformation of the following type:

f(r) =

∫ ∞

0

krJn(krr)f (kr) dkr (3.108)

This inverse Hankel transformation is evaluated in a similar way as the inverseFourier transformation in the two-dimensional case. In the three-dimensional case,

Numerical Hankel transformations 77

the Green’s displacements exhibit a first order singularity, while the Green’s strainsand stresses exhibit a second order singularity as the source-receiver distance tendsto zero. The function f(kr) is therefore decomposed into a singular part, whichis transformed analytically, and a regular part, which is transformed numerically.For the Green’s displacements, the singular part g(kr) of the function f(kr) hasthe same behaviour as the displacement field in a homogeneous fullspace due to astatic point load [36]:

g(kr) =(

α11

kr+ α2|z − z′|

)

e−kr|z−z′| (3.109)

For the Green’s strains and stresses, the singular part h(kr) of the function f(kr)has the same behaviour as the stress field in a homogeneous fullspace due to astatic point load [36]:

h(kr) =(

α1 + α2kr|z − z′|)

e−kr |z−z′| (3.110)

The coefficients α1 and α2 are determined by fitting the singular functions g(kr)and h(kr) to the function f(kr) in the high wavenumber range. The relevant inverseHankel transformations of the singular functions g(kr) and h(kr) are subsequentlyobtained as:

H−10

[

g(kr); r]

= α11

R+ α2

cos(φ)|z − z′|R2

(3.111)

H−11

[

g(kr); r]

= α1

tan(

φ2

)

R+ α2

sin(φ)|z − z′|R2

(3.112)

H−12

[

g(kr); r]

= α1

tan2(

φ2

)

R+ α2

(2 + cos(φ)) tan2(

φ2

)

|z − z′|R2

(3.113)

H−10

[

h(kr); r]

= α1cos(φ)

R2+ α2

(1 + 3 cos(2φ))|z − z′|R3

(3.114)

H−11

[

h(kr); r]

= α1sin(φ)

R2+ α2

3 cos(φ) sin(φ)|z − z′|R3

(3.115)

H−12

[

h(kr); r]

= α1

(2 + cos(φ)) tan2(

φ2

)

R2+ α2

3 sin2(φ)|z − z′|R3

(3.116)

where R =√

r2 + |z − z′|2 and φ = cos−1(|z − z′|/R).

The regular part of the function f(kr) is transformed by means of the Hankeltransformation method developed by Talman [212]. A change of variables is

78 Green’s functions of layered soils

performed in equation (3.108) using r = r0ev and kr = k0e

−w:

f(r0ev) =

∫ ∞

−∞

k0e−wJn(k0r0e

v−w)f(k0e−w)k0e

−w dw (3.117)

= k20e

−av

∫ ∞

−∞

Jn(k0r0ev−w)ea(v−w)f(k0e

−w)e−(2−a)w dw (3.118)

where k0 = 1/r0 and r0 =√rminrmax, as in the two-dimensional case. The factor

e−aveav is introduced in equation (3.118) to obtain absolutely integrable functionsin the following. Equation (3.118) is reformulated as:

f(r0ew) = k2

0e−awχ(w) (3.119)

where the function χ(w) is defined as the following convolution:

χ(w) = ϕ(w) ∗ ψ(w) (3.120)

The functions ϕ(w) and ψ(w) are defined as:

ϕ(w) = f(k0e−w)e−(2−a)w (3.121)

ψ(w) = Jn(k0r0ew)ea(w) (3.122)

The convolution theorem is used to evaluate equation (3.120) in a similar way asin the two-dimensional case. The Fourier transformation Ψ(s) of the kernel ψ(w)is given by:

Ψ(s) =1

2

(

k0r02

)is−a Γ(

n+a−is2

)

Γ(

2+n−a+is2

) (3.123)

This transformation is only possible if the function ψ(w) is absolutely integrable,which is the case if −n < a < 1.5 [1, 212]. In the present work, a value a = 1 isused. Equation (3.123) is reformulated to improve numerical stability:

Ψ(s) =1

2

(

k0r02

)is−a

ecr(s)+ici(s) (3.124)

where:

cr(s) = Re[

ln(

Γ(

n+a−is2

))]

− Re[

ln(

Γ(

2+n−a+is2

))]

(3.125)

ci(s) = Im[

ln(

Γ(

n+a−is2

))]

− Im[

ln(

Γ(

2+n−a+is2

))]

(3.126)

As the kernel Ψ(s) is not band-limited, the results of the FFT would be affectedby the Gibbs phenomenon, as in the two-dimensional case. Therefore, the function

Numerical Hankel transformations 79

−10 −5 0 5 10−15

−10

−5

0

5

10

15

w

ψ(w

)

Figure 3.3: Kernel ψ(w) of the logarithmic Hankel transformation.

ϕ(w) is multiplied with the window shown in figure 3.1a and the kernel Ψ(s) ismultiplied with the filter shown in figure 3.1b.

The filtered kernel ψ(w) is shown in figure 3.3. The function ψ(w) is calculatedfrom Ψ(s) by means of an inverse FFT using 8192 samples and a sampling interval∆w = 0.009217. The filtered kernel ψ(w) only (significantly) differs from zero

in a bounded window [wψmin, wψmax]. Using the asymptotic forms [1] of the Bessel

function in equation (3.122) for small and large arguments, the bounds of thiswindow are estimated as:

wψmin =1

a+ n

(

ln(ε) + ln(Γ(n+ 1)) − n ln

(

k0r02

))

(3.127)

wψmax = ln

(

π

γk0r0∆w

)

(3.128)

The threshold value ε and the factor γ play the same role as in equations (3.103)and (3.104) for the two-dimensional case, and the same values ε = 10−6 andγ = 0.5 are used. Using equations (3.127) and (3.128), the wavenumber range[kmin, kmax] where the function f(kr) must be sampled, can be expressed in termsof the spatial interval [rmin, rmax] of interest:

kmin =k0r0rmax

(

εΓ(n+ 1)

(

2

k0r0

)n) 1a+n

(3.129)

kmax =π

rminγ∆w(3.130)

As in the two-dimensional case, the wavenumber domain Green’s functions arecalculated in a smaller wavenumber range and extrapolated up to kmin and kmax

in the present work.Within the wavenumber range [kmin, kmax], the function f(kx) is logarithmically

sampled according to equation (3.99). The sampling interval ∆w can be

80 Green’s functions of layered soils

determined by means of the rule of thumb (3.107) used in the two-dimensionalcase.

3.6 Numerical example

As an example, the three-dimensional Green’s functions uGzj(z

′ = 0, r, θ = 0, z, ω) ofa soil are computed in this section. These functions represent the soil displacementsdue to a vertical harmonic point load at the soil’s surface.

The soil is modelled as a homogeneous halfspace with a shear modulus µ =40.5 MPa, a Poisson’s ratio ν = 0.33, a hysteretic material damping ratio β =0.03 for both dilatational and shear waves, and a density ρ = 1800 kg/m3. Thecorresponding dilatational and shear wave velocities are Cp = 300 m/s and Cs =150 m/s.

10−3

10−2

10−1

100

101

102

103

−2

−1

0

1

2x 10

−7

Wavenumber [ − ]

Dis

plac

emen

t [m

/N]

Figure 3.4: Real (black line) and imaginary (gray line) part of the wavenumberdomain Green’s function uG

zz(z′ = 0, kr, θ = 0, z = 0, ω) at 25 Hz.

The direct stiffness formulation is used to calculate the wavenumber domainGreen’s function uG

zz(z′ = 0, kr, θ = 0, z, ω). This function is calculated for

1500 dimensionless wavenumbers kr between kminr = 10−3 and kmax

r = 103.The dimensionless wavenumber kr is defined as kr = krCref/ω with a referencevelocity Cref = 150 m/s corresponding to the shear wave velocity Cs of thehalfspace. Figure 3.4 shows the wavenumber content of the Green’s functionuGzz(z

′ = 0, kr, θ = 0, z = 0, ω) at 25 Hz. Wave propagation in the verticaldirection is determined by the dispersion relations (2.72) and (2.79). For horizontalwavenumbers 0 ≤ kr ≤ s = Cs/Cp = 0.5, both vertical wavenumbers kzp and kzsare close to the real axis and both P-waves and S-waves propagate in the verticaldirection. When the horizontal wavenumber kr > s = 0.5, the vertical wavenumberkzp shifts to a value close to the imaginary axis and the P-waves become evanescentin the vertical direction. Analogously, when the horizontal wavenumber kr > 1,the vertical wavenumber kzs shifts to a value close to the imaginary axis and theS-waves become evanescent in the vertical direction. Both events are visible as dips

Numerical example 81

in the wavenumber content of the Green’s function uGzz(z

′ = 0, kr, θ = 0, z = 0, ω)corresponding to values kr = 0.5 and kr = 1. The latter is not visible due tothe sharp peak at kr = 1.073, which corresponds to the Rayleigh wave. Thedetermination of a wavenumber sampling interval according to equation (3.107)ensures that this peak is well represented.

(a)10

−210

−110

010

110

210

3−2

−1

0

1

2x 10

−7

Wavenumber [rad/m]

Dis

plac

emen

t [m

/N]

(b)10

−210

−110

010

110

210

3−2

−1

0

1

2x 10

−7

Wavenumber [rad/m]

Dis

plac

emen

t [m

/N]

Figure 3.5: Real (black line) and imaginary (gray line) part of the (a) regularand (b) singular component of the wavenumber domain Green’s function uG

zz(z′ =

0, kr, θ = 0, z = 0, ω) at 25 Hz.

The space domain Green’s function uGzz(z

′ = 0, r, θ = 0, z, ω) is calculated fromthe wavenumber domain Green’s function uG

zz(z′ = 0, kr, θ = 0, z, ω) according to

equation (3.81). To this end, the wavenumber domain Green’s function uGzz(z

′ =0, kr, θ = 0, z, ω) is decomposed into a regular and a singular part as indicated inthe previous section. Both parts are shown in figure 3.5.

The regular part of the wavenumber domain Green’s function is transformed tothe space domain by means of Talman’s algorithm. First, the function ϕ(w) iscalculated according to equation (3.121). The result is shown in figure 3.6. This

function is extrapolated with a constant value up to the bounds wψmin and wψmax

given by equations (3.127) and (3.128).Second, the convolution χ(w) of the function ϕ(w) and the kernel ψ(w) shown

in figure 3.3 is computed by means of a forward FFT, followed by a multiplicationwith the function Ψ(s), and an inverse FFT. The result χ(w) is shown in figure3.7.

Third, the function χ(w) is introduced in equation (3.119) to obtain the regularpart of the space domain Green’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω). Its

singular part is obtained analytically from equation (3.111). Both parts are shownin figure 3.8.

The effect of the window applied to the function ϕ(w) and the filter applied tothe kernel Ψ(s) is illustrated in figure 3.9. This figure shows the regular part of thespace domain Green’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω) calculated with the

original algorithm proposed by Talman [212], without window or filter. At smalldistances r, the function uG

zz(z′ = 0, r, θ = 0, z = 0, ω) exhibits sharp oscillations

82 Green’s functions of layered soils

−10 −5 0 5 10−2

−1

0

1

2x 10

−7

w

φ(w

)

Figure 3.6: Real (black line) and imaginary (gray line) part of the function ϕ(w).

−10 −5 0 5 10−6

−4

−2

0

2

4

6x 10

−8

w

χ(w

)

Figure 3.7: Real (black line) and imaginary (gray line) part of the function χ(w).

(a)10

−310

−210

−110

010

110

210

3−1

−0.5

0

0.5

1x 10

−8

Distance [m]

Dis

plac

emen

t [m

/N]

(b)10

−310

−210

−110

010

110

210

3−1

−0.5

0

0.5

1x 10

−8

Distance [m]

Dis

plac

emen

t [m

/N]

Figure 3.8: Real (black line) and imaginary (gray line) part of the (a) regular and(b) singular component of the space domain Green’s function uG

zz(z′ = 0, r, θ =

0, z = 0, ω) at 25 Hz.

due to the Gibbs phenomenon. These oscillations are problematic if the Green’sfunctions are used in a boundary element formulation, where they are evaluated

Numerical example 83

at small source-receiver distances to account for the interaction between adjacentpoints on the boundary.

10−3

10−2

10−1

100

101

102

103

−1

−0.5

0

0.5

1x 10

−8

Distance [m]

Dis

plac

emen

t [m

/N]

Figure 3.9: Real (black line) and imaginary (gray line) part of the regularcomponent of the space domain Green’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω)

at 25 Hz, calculated without window or filter.

Finally, the regular and the singular part are combined to obtain the spacedomain Green’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω), which is shown in figure

3.10.

10−3

10−2

10−1

100

101

102

103

−1

−0.5

0

0.5

1x 10

−8

Distance [m]

Dis

plac

emen

t [m

/N]

Figure 3.10: Real (black line) and imaginary (gray line) part of the space domainGreen’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω) at 25 Hz .

In a similar way, the space domain Green’s function uGzr(z

′ = 0, r, θ = 0, z, ω)is calculated from the wavenumber domain Green’s function uG

zr(z′ = 0, kr, θ =

0, z, ω) according to equation (3.79). Figure 3.11 shows the real part of bothGreen’s functions uG

zj(z′ = 0, r, θ = 0, z, ω) at 25 Hz on a grid of receivers in

the (r, z)-plane. Along the axis of symmetry r = 0, the motion consists of adilatational wave. A shear window is travelling in an average direction of 45. Ina region adjacent to the surface, the Rayleigh wave is clearly visible.

84 Green’s functions of layered soils

0 5 10 15 20 25

0

5

10

15

20

Radial distance [m]

Dep

th [m

]

Figure 3.11: Real part of the Green’s functions uGzj(z

′ = 0, r, θ = 0, z, ω) at 25 Hz.

Figure 3.12 shows the frequency content of the Green’s function uGzz(z

′ =0, r, θ = 0, z = 0, ω) at four different receiver locations. The oscillation of theresponse uG

zz(z′ = 0, r, θ = 0, z = 0, ω) as a function of the frequency is due to

the interference of P-waves, S-waves, and Rayleigh waves [185]. The responseuGzz(z

′ = 0, r, θ = 0, z = 0, ω) decreases as the source-receiver distance r increasesdue to geometrical spreading and material damping. While the attenuation due togeometrical spreading is frequency independent, the attenuation due to hystereticmaterial damping is proportional to the ratio of the wavelength and the source-receiver distance. As a result, a stronger attenuation is observed in the highfrequency range.

3.7 Conclusion

In this chapter, the calculation of the Green’s displacement and stress functions ofa soil is considered. These functions represent the dynamic response of a soil dueto a unit load. The Green’s functions serve as the basis of the boundary integralformulation, which is used in the next chapter to model dynamic soil-structureinteraction.

The Green’s functions are calculated in the frequency-wavenumber domain, bymeans of the direct stiffness method. They are subsequently transformed to the

Conclusion 85

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 3.12: Modulus of the Green’s function uGzz(z

′ = 0, r, θ = 0, z = 0, ω) for asource-receiver distance r of (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m.

frequency-space domain. Inverse Fourier and Hankel transformations are used toobtain the two-dimensional and three-dimensional Green’s functions, respectively.The transformations are performed by means of an algorithm developed by Talman[212], using a logarithmic sampling scheme. The use of a logarithmic samplingscheme is advantageous for problems involving different length scales, such asdynamic soil-structure interaction problems, where both the response at adjacentpoints on the soil-structure interface and the far field response are of interest. Theoriginal algorithm proposed by Talman is improved through the use of a windowand a filter to mitigate artifacts in the results caused by the Gibbs phenomenon.

86 Green’s functions of layered soils

Chapter 4

Dynamic soil-structure

interaction

4.1 Introduction

Vibrations in structures are caused by external forces applied to the structure or bywaves that propagate through the soil and impinge on its foundation. These wavesmay be due to an earthquake, traffic, industrial activities, and construction works.The response of the structure can be calculated using a subdomain formulation fordynamic soil-structure interaction [16, 36]. Models based on this formulation havebeen used to predict vibrations caused by road traffic [71, 135, 168, 169], trains[136], subways [39, 47, 90], and pile driving [143].

In the present work, a similar method is used to simulate the foundation-soiltransfer functions measured in Lincent (section 1.4). The foundation is modelledwith the finite element method and the soil is modelled with the boundary elementmethod. The boundary element method is based on the Green’s functions of alayered soil, which are calculated with the direct stiffness method as described inthe previous chapters.

This chapter briefly reviews the subdomain formulation for dynamic soil-structure interaction. In section 4.2, the equations of motion for the coupled soil-structure system are formulated. These equations are governed by the impedanceof the soil and the structure. A boundary integral formulation is used to calculatethe soil impedance. The boundary integral equations are solved numerically bymeans of the boundary element method. The boundary integral formulation isdiscussed in section 4.3. In section 4.4, the subdomain formulation is used tosimulate the foundation-soil transfer functions measured in Lincent.

87

88 Dynamic soil-structure interaction

4.2 Subdomain formulation

In this section, the subdomain formulation for dynamic soil-structure interaction[16, 36] is outlined. This formulation is applicable to both surface and embeddedfoundations, and allows to account for both external forces on the structure andincident wave fields due to earthquakes or other distant sources. In the presentwork, however, the focus is restricted to surface foundations subjected to externalforces.

ΓsσΓsσ

Ωs

Σ

Γs∞

Ωb

Γbσ

Figure 4.1: Geometry of the subdomains.

The dynamic interaction problem is decomposed into two subdomains (figure4.1): the structure Ωb and the soil domain Ωs. The interaction problem is solvedby enforcing continuity of displacements and equilibrium of stresses on the interfaceΣ between both subdomains.

The displacements of the soil and the structure at a position x and at circularfrequency ω are denoted by us(x, ω) and ub(x, ω), respectively. In the following, allequations are formulated in the frequency domain and the hats above the variablesare omitted.

4.2.1 The structural subdomain

The structural subdomain Ωb is bounded by the surface Γbσ and the soil-structureinterface Σ. The unit outward normal vector on the boundary of the structuraldomain is denoted by nb. On the surface Γbσ, the tractions t

nb

b are imposed.On the soil-structure interface, stress equilibrium and displacement continuityis enforced. The displacement field ub of the structure satisfies the following

Subdomain formulation 89

equilibrium equation and boundary conditions:

∇ · σb(ub) + ρbb + ρbω2ub = 0 in Ωb (4.1)

tnb

b (ub) − tnb

b = 0 on Γbσ (4.2)

tnb

b (ub) + tnss (us) = 0 on Σ (4.3)

ub − us = 0 on Σ (4.4)

where ∇ · σb(ub) is the divergence of the Cauchy stress tensor σb(ub) due to thedisplacement field ub, tnb

b (ub) is the traction vector due to the displacement fieldub on a plane with unit outward normal vector nb, ρb is the density, and ρbb isthe body force vector.

4.2.2 The soil subdomain

The semi-infinite soil domain Ωs is bounded by the surface Γsσ, the soil-structureinterface Σ, and the outer boundary Γs∞. The unit outward normal vector on theboundary of the soil domain is denoted by ns.

The radiated wave field us in the soil is due to the motion of the interface Σ. Thiswave field satisfies the following equilibrium equation and boundary conditions:

∇ · σs(us) + ρsω2us = 0 in Ωs (4.5)

tnss (us) = 0 on Γsσ (4.6)

tnss (us) + tnb

b (ub) = 0 on Σ (4.7)

us − ub = 0 on Σ (4.8)

R(us) = 0 on Γs∞ (4.9)

The operator R(us) in equation (4.9) denotes Sommerfeld’s radiation conditionson the displacements us on the outer boundary Γs∞ of the unbounded domain Ωs.Sommerfeld’s radiation conditions have been discussed in section 2.2.

4.2.3 The interaction problem

The dynamic soil-structure interaction problem is defined by equations (4.1–4.9).In order to obtain an approximate solution of the interaction problem, the radiatedwave field us and the structural displacements ub are projected on a kinematicbasis consisting of Q basis vectors, denoted by ψbm in the structural domain and

90 Dynamic soil-structure interaction

by ψsm in the soil domain:

ub =

Q∑

m=1

αmψbm (4.10)

us =

Q∑

m=1

αmψsm (4.11)

The eigenmodes of the structure with free boundary conditions are commonly usedas basis vectors ψbm in the structural domain. The basis vectors ψsm in the soildomain are the wave fields radiated by the structural eigenmodes ψbm and satisfythe following equations:

∇ · σs(ψsm) + ρsω2ψsm = 0 in Ωs (4.12)

tnss (ψsm) = 0 on Γsσ (4.13)

R(ψsm) = 0 on Γs∞ (4.14)

ψsm −ψbm = 0 on Σ (4.15)

The unknown modal coordinates αm in equations (4.10) and (4.11) follow fromthe principle of virtual work applied to the structural domain:

∫

Ωb

ε(v) : σb(ub) dΩ − ω2

∫

Ωb

v · ρbub dΩ

=

∫

Σ

v · tnb

b (ub) dΣ +

∫

Γbσ

v · tnb

b dΓ +

∫

Ωb

v · ρbb dΩ (4.16)

where v denotes any virtual displacement field and ε(v) denotes the correspondingvirtual strain field. Accounting for the stress equilibrium (4.3) on the soil-structureinterface Σ, the equilibrium equation (4.16) becomes:

∫

Ωb

ε(v) : σb(ub) dΩ − ω2

∫

Ωb

v · ρbub dΩ +

∫

Σ

v · tnss (us) dΣ

=

∫

Γbσ

v · tnb

b dΓ +

∫

Ωb

v · ρbb dΩ (4.17)

The decompositions (4.10) and (4.11) are introduced into equation (4.17). In astandard Galerkin procedure, the virtual displacement field v is decomposed in asimilar way as v =

∑Qn=1 ψbnδαn, where δαn are the virtual modal coordinates.

This results into the following system of equations:[

Kb − ω2Mb + Ks

]

α = fb (4.18)

Boundary integral formulation 91

where α is a vector collecting the modal coordinates αm. The modal stiffness andmass matrices Kb and Mb are equal to:

[Kb]nm =

∫

Ωb

ε(ψbn) : σb(ψbm) dΩ (4.19)

[Mb]nm =

∫

Ωb

ψbn · ρbψbm dΩ (4.20)

The impedance matrix Ks of the soil is equal to:

[Ks]nm =

∫

Σ

ψbn · tnss (ψsm) dΣ (4.21)

The vector fb due to the external forces on the structure is defined as:

[fb]n =

∫

Γbσ

ψbn · tnb

b dΓ +

∫

Ωb

ψbn · ρbb dΩ (4.22)

The stiffness matrix Kb, the mass matrix Mb, and the force vector fb in equations(4.19), (4.20), and (4.22) are obtained by means of a finite element formulation.The soil impedance matrix Ks in equation (4.21) is calculated by means of aboundary integral formulation.

4.3 Boundary integral formulation

In this section, the boundary integral equations are formulated. These equationsare used to calculate the tractions tns

s (ψsm) on the soil-structure interface Σand the radiated wave fields ψsm in the soil domain Ωs due to the (known)displacements ψsm on the interface Σ. The tractions tns

s (ψsm) are needed tocompute the soil impedance matrix Ks from equation (4.21). The radiated wavefields ψsm allow to compute the free field response from equation (4.11), using themodal coordinates αm obtained from equation (4.18).

4.3.1 The dynamic reciprocity theorem

The dynamic reciprocity theorem is the basis of all boundary integral equations inelastodynamics [225]. This theorem is an extension of Betti’s classical reciprocitytheorem of elastostatics, and relates the displacements and tractions in twoelastodynamics states.

Consider two elastodynamic states of a volume Ω with a boundary Γ and a unitoutward normal vector n, characterized by the body forces ρb1 and ρb2 and thetractions tn1 and tn2 , resulting in the displacements u1 and u2. In the frequencydomain, the dynamic reciprocity theorem for these elastodynamic states reads as

92 Dynamic soil-structure interaction

follows:

∫

Γ

tn1j(x)u2j(x) dΓ +

∫

Ω

ρb1j(x)u2j(x) dΩ

=

∫

Γ

tn2j(x)u1j(x) dΓ +

∫

Ω

ρb2j(x)u1j(x) dΩ (4.23)

where the summation convention is used.

4.3.2 The boundary integral equations

The dynamic reciprocity theorem is used to formulate an integral equation thatexpresses the relation between the field variables in a known elastodynamic stateand an unknown elastodynamic state of the soil domain Ω with the boundary Γand the unit outward normal vector n.

The known state is the fundamental singular solution, where a concentratedpulse ρbj(x) = δ(x−x′)δij is applied at the point x′ of Ω in the direction ei. Theresulting displacements in Ω and tractions on Γ are the three-dimensional Green’sfunctions uG

ij(x′,x) and tGn

ij (x′,x) of the semi-infinite soil domain Ωs.The unknown state is characterized by the displacements u(x) and tractions

tn(x). In this state, the body forces ρb(x) are assumed to be zero.From the dynamic reciprocity theorem (4.23) follows that:

ui(x′) =

∫

Γ

uGij(x

′,x)tnj (x) dΓ −∫

Γ

tGnij (x′,x)uj(x) dΓ (4.24)

The radiated wave field ψsm is considered as unknown state and the boundaryintegral equation (4.24) is applied to the entire soil domain Ωs with the boundaryΣ ∪ Γsσ ∪ Γs∞:

ψsmi(x′) =

∫

Σ

uGij(x

′,x)tns

sj (ψsm)(x) dΓ −∫

Σ

tGns

ij (x′,x)ψsmj(x) dΓ

+

∫

Γsσ

uGij(x

′,x)tns

sj (ψsm)(x) dΓ −∫

Γsσ

tGns

ij (x′,x)ψsmj(x) dΓ

+

∫

Γs∞

uGij(x

′,x)tns

sj (ψsm)(x) dΓ −∫

Γs∞

tGns

ij (x′,x)ψsmj(x) dΓ (4.25)

In equation (4.25), the tractions tns

sj (ψsm)(x) and tGns

ij (x′,x) are zero on the surfaceΓsσ and the integrals on the outer boundary Γs∞ vanish due to Sommerfeld’sradiation conditions. As a result, equation (4.25) reduces to:

ψsmi(x′) =

∫

Σ

uGij(x

′,x)tns

sj (ψsm)(x) dΓ −∫

Σ

tGns

ij (x′,x)ψsmj(x) dΓ (4.26)

Numerical example 93

This boundary integral equation is also valid when an embedded foundation isconsidered. In the present case, the soil-structure interface Σ is located at thesoil’s surface, where the Green’s functions tGns

ij (x′,x) are zero. Equation (4.26)therefore reduces further to:

ψsmi(x′) =

∫

Σ

uGij(x

′,x)tns

sj (ψsm)(x) dΓ (4.27)

This equation establishes a relation between the known displacement ψsmi(x′) at

a point x′ on the interface Σ and the unknown tractions tnss (ψsm) along the soil-

structure interface Σ. The tractions can be calculated by means of a boundaryelement formulation [16, 25, 54, 165] through a discretization of the field variableson the interface Σ. In the present work, constant element-based shape functions areused and the integral equation is solved with a collocation scheme. Subsequently,equation (4.27) is used to compute the radiated wave field ψsmi(x

′) at a point x′

in the soil domain Ωs from the tractions tnss (ψsm) on the interface Σ.

4.4 Numerical example

The subdomain formulation is now applied to the dynamic soil-structure inter-action problem introduced in section 1.4. The aim is to predict the foundation-soil transfer functions H(ω) at 4 m, 8 m, 16 m, and 32 m from the center of thefoundation. The frequency range of interest extends to 150 Hz.

The soil is modelled as a homogeneous halfspace with the same properties as insection 3.6. The foundation is modelled as a square block of reinforced concretewith a width of 0.5 m and a height of 0.2 m. The Young’s modulus of reinforcedconcrete is 35 GPa, the Poisson’s coefficient is 0.2 and the density is 2500 kg/m3.

A MATLAB toolbox for boundary element calculations developed at theStructural Mechanics division of K.U.Leuven is used to calculate the impedanceof the soil and the radiated wave field. The boundary element method is based onthe Green’s functions of the soil, which are calculated with the MATLAB toolboxEDT 2.0 as described in the previous chapters.

4.4.1 The structural modes

First, a modal analysis of the concrete foundation is performed. The foundationis modelled by means of the finite element method using 25 × 25 × 10 eight-nodebrick elements. The eigenvalue problem is solved with the Lanczos algorithm.The lowest eigenfrequencies of the non-rigid body modes are equal to 1452 Hz,2084 Hz, and 2357 Hz. The convergence of the modal analysis is verified witha model consisting of 50 × 50 × 20 elements of the same type. This leads toeigenfrequencies of 1450 Hz, 2077 Hz, and 2348 Hz, which are very close to thevalues obtained with the coarser model.

The first nine eigenmodes (including six rigid body modes) are shown in figure4.2. Only the modes in figures 4.2c and 4.2i are not orthogonal to the (vertical)

94 Dynamic soil-structure interaction

impact force on the foundation. The eigenfrequency corresponding to the bendingmode shown in figure 4.2i equals 2357 Hz, which is far beyond the frequency rangeof interest. Therefore, only the vertical rigid body mode shown in figure 4.2c isused in the analysis. This mode is denoted as ψb1.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.2: First nine eigenmodes of the concrete foundation. Modes (a) to (f) arerigid body modes. The eigenfrequencies corresponding to the other modes are (g)1452 Hz, (h) 2084 Hz, and (i) 2357 Hz.

4.4.2 The modal soil tractions

The eigenmode ψb1 gives rise to a radiated wave field ψs1 in the soil and modal soiltractions tns

s (ψs1) on the soil-structure interface. These tractions are calculatedfrom equation (4.27) using the Green’s displacement functions uG

i (x′,x) of the soil.The Green’s displacement functions uG

i (x′,x) are calculated in the same way andusing the same parameter values as in subsection 3.6. The soil-structure interfaceis discretized using 10 × 10 boundary elements with constant shape functions.Over each element, the boundary integral equation is calculated with a Gaussianquadrature scheme, using 36 Gaussian points for the regular integrals and 144Gaussian points for the singular integrals.

The resulting modal soil tractions tnss (ψs1) are shown in figure 4.3 at a frequency

of 100 Hz. Both the vertical and horizontal tractions exhibit a singular behaviournear the edges of the foundation. Figure 4.3c shows that the vertical tractions onthe soil are not in phase with the motion of the foundation. The imaginary part ofthe tractions is due to energy dissipation through material and radiation dampingin the soil.

Numerical example 95

(a) (b) (c)

Figure 4.3: Real (top) and imaginary (bottom) part of the modal soil tractions(a) tns

sx(ψs1), (b) tnssy (ψs1), and (c) tns

sz (ψs1) at 100 Hz.

4.4.3 The foundation-soil impedance

In the present analysis, the modal foundation-soil impedance matrix only containsa single element per frequency. This matrix is composed of the modal foundationimpedance Kb − ω2Mb and the modal soil impedance Ks. The foundationimpedance is given by:

[

Kb − ω2Mb

]

11= −ω2mb (4.28)

where mb = 125 kg is the mass of the foundation. The soil impedance is obtainedfrom the modal soil tractions tns

s (ψs1) according to equation (4.21). The integralis computed by means of a Gaussian quadrature scheme.

(a)0 25 50 75 100 125 150

−1

−0.5

0

0.5

1x 10

8

Frequency [Hz]

Impe

danc

e [N

/m]

(b)0 25 50 75 100 125 150

−1

−0.5

0

0.5

1x 10

8

Frequency [Hz]

Impe

danc

e [N

/m]

Figure 4.4: Real (black line) and imaginary (gray line) part of (a) the structuralimpedance

[

Kb − ω2Mb

]

11and (b) the soil impedance [Ks]11.

96 Dynamic soil-structure interaction

The components[

Kb − ω2Mb

]

11and [Ks]11 of the foundation-soil impedance

are shown in figure 4.4. The soil impedance in figure 4.4b corresponds well to thereference impedance curves for rectangular surface foundations given by Sieffertand Cevaer [196]. At low frequencies, the dimensions of the foundation are smallcompared to the wavelength of the waves in the soil and a vertical vibration of thefoundation generates longitudinal, shear, and surface waves. At high frequencies,however, each point on the surface behaves as an independent source that radiatesone-dimensional waves perpendicular to the surface. Oblique waves interfere andvanish [53]. As a result, the soil impedance directly follows from the stiffnessmatrix of the halfspace element used in the direct stiffness method for a horizontalwavenumber kx = 0:

[Ks]11 = iωρCpA for ω → ∞ (4.29)

where A = 0.25 m2 is the area of the foundation. Equation (4.29) explains thelinear increase of the imaginary part of the soil impedance with the frequency infigure 4.4b.

0 25 50 75 100 125 1500

0.5

1

1.5

2x 10

8

Frequency [Hz]

Impe

danc

e [N

/m]

Figure 4.5: Modulus of the foundation-soil impedance[

Kb − ω2Mb + Ks

]

11.

Figure 4.5 shows the modulus of the foundation-soil impedance[

Kb − ω2Mb + Ks

]

11. In the low frequency range, the foundation impedance

is small and the soil impedance dominates. As the frequency increases, thefoundation impedance increases quadratically, while the soil impedance increasesonly linearly. Consequently, the foundation impedance becomes dominant in thehigh frequency range. Around 75 Hz, the foundation-soil impedance reaches a(not so pronounced) minimum.

4.4.4 The radiated wave field

The modal soil tractions tnss (ψs1) are used to calculate the radiated wave field ψs1

according to equation (4.27), using a Gaussian quadrature scheme. The radiatedwave field ψs1 is shown in figure 4.6 at a frequency of 100 Hz. The Rayleigh wave,which dominates the motion at the soil’s surface, is clearly visible.

Numerical example 97

(a)

(b)

Figure 4.6: (a) Real and (b) imaginary part of the radiated wave field ψs1 at100 Hz.

4.4.5 The foundation-soil transfer functions

Finally, equation (4.18) is used to calculate the modal coordinate α1. Thefoundation-soil transfer function is subsequently obtained as the product of themodal coordinate α1 and the radiated wave field ψs1. Figure 4.7 compares theresults with the Green’s function uG

zz(z′ = 0, r, θ = 0, z = 0, ω) calculated in

section 3.6.In the near field, the transfer function H(ω) reaches a maximum around 75 Hz,

where the foundation-soil impedance is minimal. In the far field, the transferfunction is smaller due to geometric and material damping. As the attenuationdue to material damping is stronger in the high frequency range, the maximum ofthe transfer function H(ω) shifts to a lower frequency.

In the low frequency range, the wavelength of the waves in the soil is largecompared to the dimensions of the foundation. The impact of the presence ofthe foundation on the wave field in the soil is limited. Moreover, the foundation

98 Dynamic soil-structure interaction

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 4.7: Modulus of the foundation-soil transfer function H(ω) (black line) andthe Green’s function uG

zz (gray line) for different receivers, located at (a) 4 m, (b)8 m, (c) 16 m, and (d) 32 m from the center of the foundation.

impedance is small compared to the soil impedance. As a result, the effect ofdynamic foundation-soil interaction is limited and the transfer functions are verysimilar to the Green’s functions. In the high frequency range, the wavelengthis smaller compared to the dimensions of the foundation. The presence of thefoundation consequently disturbs the wave field in the soil. Furthermore, thefoundation impedance is larger than in the low frequency range. As a result, theforce transmitted to the soil is reduced and the transfer functions are smaller thanthe Green’s functions.

The convergence of the simulation is determined by the wavenumber samplingof the Green’s functions and the number of boundary elements. In order toverify the convergence, an additional simulation is performed using a refinedmodel. In this model, the Green’s functions of the soil are calculated for 6000dimensionless wavenumbers kr between kmin

r = 10−6 and kmaxr = 106 instead of

1500 dimensionless wavenumbers kr between kminr = 10−3 and kmax

r = 103 andthe number of boundary elements is increased from 10× 10 to 20× 20. Figure 4.8compares the results obtained with the original model and the refined model. Thecorrespondence is satisfactory.

Conclusion 99

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 4.8: Modulus of the foundation-soil transfer function H(ω) for differentreceivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of thefoundation. The gray line is calculated with a refined model.

4.5 Conclusion

In this chapter, the subdomain formulation for dynamic soil-structure interactiondeveloped by Aubry and Clouteau [16, 36] is briefly reviewed. The elastodynamicequations and the boundary conditions for both the structural and the soilsubdomain are formulated. The displacement field is projected on a kinematicbasis consisting of the eigenmodes of the structure. The equilibrium equations andthe boundary conditions for the structural domain are enforced in the weak sense,yielding a system of algebraic equations for the modal coordinates. This system ofequations depends on the modal stiffness and mass matrices of the structure andthe impedance of the soil. The stiffness and mass matrices of the structure arecalculated with the finite element method. The soil impedance and the radiatedwave field in the soil are calculated with a boundary integral formulation.

The subdomain formulation is applied to simulate the foundation-soil transferfunctions measured in Lincent. No comparison with the experimental data ismade as the simulation is based on a synthetic soil profile instead of the actualsoil profile at the site in Lincent. In the next chapter, the SASW method isused to update the soil profile in the dynamic soil-structure interaction model.

100 Dynamic soil-structure interaction

The foundation-soil transfer functions are subsequently recalculated and comparedwith the experimental data.

Chapter 5

The SASW method

5.1 Introduction

The SASW method is a non-invasive method to determine the dynamic shearmodulus of shallow soil layers [151, 231]. The method has been used to investigatepavement systems [152], to assess the quality of ground improvement [41], todetermine the thickness of waste deposits [114], and to identify the dynamic soilproperties for the prediction of ground vibrations [136, 143, 169]. Within theframe of the present work, the SASW method is applied to identify the dynamicsoil properties at the site in Lincent considered in section 1.4.

The SASW method is based on the dispersive characteristics of surface wavesin a layered medium and consists of three steps. The first step involves an insitu experiment where vibrations are generated at the soil’s surface using a fallingweight, an impact hammer, or a hydraulic shaker. The free field response ismeasured with geophones or accelerometers up to a distance of typically 50 m.In the second step, an experimental dispersion curve CE

R(ω) is determined usingthe phase of the transfer functions between the receiver signals. It is assumedthat the response at a sufficiently large distance from the source is dominated bydispersive surface waves. In the third step, an inverse problem is solved to obtainthe dynamic shear modulus of the soil. The direct stiffness method [113] or anequivalent formulation is used to calculate the theoretical dispersion curve CT

R (ω)of a soil with a given stiffness profile. The stiffness profile is iteratively adjustedin order to minimize a misfit function that measures the distance between thetheoretical and the experimental dispersion curve. The minimization problem isusually solved with a gradient based local optimization method. As an alternative,a global optimization scheme can be used, such as the method of coupled localminimizers [49].

This chapter is organized as follows. Section 5.2 focuses on the surface waves in alayered halfspace and explains how these waves allow to determine the variation of

101

102 The SASW method

the dynamic soil properties with depth. Section 5.3 addresses Nazarian’s method[151] to determine the experimental dispersion curve from in situ measurements.Section 5.4 describes a method to calculate the theoretical dispersion curve ofa layered soil. In section 5.5, the distance between the theoretical and theexperimental dispersion curve is minimized in order to determine the dynamicshear modulus of the soil at the site in Lincent. The identified soil profile is usedin section 5.6 to recalculate the foundation-soil transfer functions computed in theprevious chapter. The results are compared with experimental data.

5.2 Surface waves in a layered halfspace

In this section, the direct stiffness formulation is used to calculate the surface wavemodes of a layered halfspace. Due to the uncoupling of dilatational and shearwaves, a distinction can be made between in-plane surface waves (or Rayleighwaves) and out-of-plane surface waves (or Love waves).

5.2.1 Rayleigh waves

The in-plane free surface waves or natural modes of vibration in a layered halfspaceare equal to the displacements uPSV when the external load vector tPSV in theequilibrium equation (2.179) equals zero. Non-trivial solutions for uPSV can beobtained if the stiffness matrix KPSV is singular or if the determinant of KPSV isequal to zero:

det KPSV = 0 (5.1)

Equation (5.1) corresponds to an eigenvalue problem in terms of the realfrequency ω and the complex horizontal wavenumber kx, whose imaginary partrepresents wave attenuation in the horizontal direction. The eigenvalue problemis transcendental, has an infinite number of solutions, and must be solved bysearch techniques. For each frequency ω, the phase velocity CR(ω) of the Rayleighwave is obtained as the ratio ω/kx where (ω, kx) is a solution of the characteristicequation (5.1). At a given frequency, multiple Rayleigh waves corresponding tomultiple solutions of the eigenvalue problem may exist. The Rayleigh wave withthe lowest phase velocity is referred to as the fundamental Rayleigh wave.

For a homogeneous halfspace with zero material damping, the characteristicequation (5.1) reduces to the classical cubic equation that was first formulated byRayleigh [172]. In this case, a single non-dispersive Rayleigh wave exists with aphase velocity CR independent of the frequency ω.

In the case of a layered soil supported by a rigid stratum, the thin layerformulation [113] can alternatively be used to assemble the stiffness matrix KPSV.This leads to a quadratic eigenvalue problem that can be reduced to a lineareigenvalue problem of the same dimension and solved by standard techniques [113].An important drawback of the thin layer approach, however, is that it is only

Surface waves in a layered halfspace 103

applicable to a layered soil supported by a rigid stratum. A hybrid formulation,where thin layer elements are coupled to a halfspace element, offers a solution, butagain leads to a transcendental eigenvalue problem.

5.2.2 Love waves

The out-of-plane free surface waves in a layered halfspace are equal to thedisplacements uSH when the external load vector tSH in the equilibrium equation(2.180) equals zero. Non-trivial solutions for uSH can be obtained if thedeterminant of KSH is equal to zero:

det KSH = 0 (5.2)

The eigenvalue problem in equation (5.2) is similar to equation (5.1) and can besolved in an analogous way to obtain the phase velocity CL(ω) of the Love waves.

5.2.3 Numerical example

As an example, the phase velocity of the Rayleigh and Love waves in a layered soilis calculated using the direct stiffness formulation. The soil profile is described intable 5.1. For each layer, the thickness d, the shear and dilatational wave velocitiesCs and Cp, and the density ρ are given. As the influence of material damping onthe phase velocity is negligible, material damping is not accounted for.

Layer d Cs Cp ρ[m] [m/s] [m/s] [kg/m3]

1 0.97 143 286 18002 1.90 168 336 18003 ∞ 259 518 1800

Table 5.1: Soil profile.

Figure 5.1 shows the phase velocity of the Rayleigh and Love waves. In the lowfrequency range, only the a single Rayleigh wave and a single Love wave exist.Higher order surface waves appear at higher frequencies, which are referred to asthe cut-off frequencies of these waves.

The surface waves are dispersive due to the variation of the soil properties withdepth. In the low frequency range, the surface wavelength is large and the surfacewave reaches deep and stiff layers, resulting in a high phase velocity. As thefrequency increases, the surface wavelength decreases and the surface wave travelsthrough shallow and soft layers, resulting in a low phase velocity. This dispersivebehaviour is the basis of the SASW method, where the dispersion curve CR(ω) ofthe fundamental Rayleigh wave is used to identify the dynamic soil properties asa function of depth.

104 The SASW method

(a)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(b)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 5.1: Phase velocity of the (a) Rayleigh and (b) Love waves in a layeredhalfspace.

At 80 Hz, three Rayleigh waves and three Love waves exist. The correspondingdisplacement fields uPSV and uSH are transformed from the wavenumber domainto the space domain according to equation (2.96). The resulting two-dimensionalwave fields are shown in figure 5.2.

5.3 The experimental dispersion curve

The experimental dispersion curve CER(ω) of the site in Lincent is determined by

means of the experiment described in section 1.4. Whereas in previous chaptersthe modulus of the foundation-soil transfer functions has been considered, theSASW method is based on the phase of the transfer functions between the freefield vibrations at different receivers.

Figure 5.3 shows the time history of the free field acceleration due to a hammerimpact on the concrete foundation as a function of the distance from the source.As all traces are scaled with respect to their peak value, the effect of attenuationis not visible. When the arrival times of the ground vibrations are compared, itis observed that the wave propagation in the soil delays the time signals for anincreasing distance from the source. Based on the ratio of the distance and thetime delay, the surface wave velocity is roughly estimated at 140 m/s.

A more accurate estimate of the surface wave velocity is obtained by means ofNazarian’s method [151] using the phase of the transfer functions between pairs ofreceivers. First, the cross power spectral density Skij(ω) between receivers i and jfor event k is computed as:

Skij(ω) =1

Taki (ω)ak∗j (ω) (5.3)

where T is the duration of the measurement, aki (ω) denotes the frequency content ofthe acceleration at receiver i for event k, and ak∗j (ω) denotes the complex conjugate

The experimental dispersion curve 105

(a) (d)

(b) (e)

(c) (f)

Figure 5.2: (a) First, (b) second, and (c) third Rayleigh wave and (d) first, (e)second, and (f) third Love wave in a layered halfspace at 80 Hz.

of akj (ω). If i = j, this function Skii(ω) is referred to as the auto power spectral

density. The average cross power spectral density Sij(ω) between receivers i andj is equal to:

Sij(ω) =1

N

N∑

k=1

Skij(ω) (5.4)

106 The SASW method

2 3 4 6 8 12 16 24 32 48

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance [m]

Tim

e [s

]

Figure 5.3: Time history of the free field acceleration due to a hammer impact onthe concrete foundation as a function of the distance from the source.

withN the number of events. A number of 10 events is used in the present analysis.Next, the transfer function Hij(ω) from receiver i to receiver j is estimated as:

Hij(ω) =Sij(ω)

Sii(ω)(5.5)

Equation (5.5) is referred to as the H1 estimator of the transfer function [64]. Thecoherence function Γij(ω) between the signals at receivers i and j is defined as:

Γij(ω) =Sij(ω)S∗

ij(ω)

Sii(ω)S∗jj(ω)

(5.6)

The coherence function Γij(ω) is a measure of the data quality. A unit valueindicates a perfectly linear relation between the signals i and j. A smallercoherence may indicate noise disturbing the measurements or non-linear behaviourof the soil.

The receivers at 2 m and 4 m, 3 m and 6 m, 4 m and 8 m, 6 m and 12 m, 8 m and16 m, 12 m and 24 m, 16 m and 32 m, and 24 m and 48 m from the center of thefoundation are taken as pairs. For each pair, the phase velocity of the surface wave

The experimental dispersion curve 107

is estimated as:

CER(ω) =

ω∆rijθij(ω)

(5.7)

where ∆rij is the distance between the receivers and θij(ω) is the unfolded phase

of the cross power spectral density Sij(ω). For a fixed frequency ω, the estimationof the dispersion curve CE

R(ω) is withheld if the following criteria are met:

Γij(ω) ≥ Γmin (5.8)

rmin ≤ ∆rijλE

R(ω)≤ rmax (5.9)

Equation (5.8) imposes a threshold on the coherence function to limit the influenceof incoherent noise. A value Γmin = 0.95 is used. Equation (5.9) ensures thatthe ratio of the distance ∆rij and the estimated surface wavelength λE

R(ω) =2πCE

R(ω)/ω is within certain bounds. The lower bound rmin acts as a high-passfilter that limits the contribution of body waves, while the upper bound rmax

serves as a low-pass filter to remove the high frequency components contaminatedby coherent noise [151]. Values rmin = 1 and rmax = 3 are used.

0 25 50 75 100 125 150 175 2000

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 5.4: Experimental dispersion curve (gray dots) and approximatingpolynomial (black dots) obtained for the site in Lincent.

Figure 5.4 shows the resulting phase velocity as well as a fifth order polynomialapproximation. The dispersion curve is obtained in the frequency range between19 Hz and 167 Hz where it decreases from 198 m/s to 132 m/s. The correspondingRayleigh wavelength varies from 10.4 m to 0.8 m. Earlier SASW tests [167]performed at different locations on the same site have lead to similar dispersioncurves.

In the following, the polynomial approximation in figure 5.4 is referred to as theexperimental dispersion curve CE

R(ω).

108 The SASW method

5.4 The theoretical dispersion curve

In the SASW method, it is customary to assume that the displacements at thesoil’s surface are dominated by the fundamental Rayleigh wave. The direct stiffnessformulation or an equivalent method is used to determine the phase velocity of thefundamental Rayleigh wave as discussed in section 5.2. The resulting theoreticaldispersion curve CT

R (ω) is compared with the experimental dispersion curve CER(ω)

in order to identify the soil profile. This approach can lead to erroneous results ifthe soil contains soft layers overlain by stiffer layers. In such cases, higher Rayleighmodes may affect the surface displacements and the dispersion curve derived fromthe experimental data may differ from the fundamental Rayleigh wave dispersioncurve [88, 216]. Several authors have attempted to tackle this problem followingtwo different strategies.

In the first strategy, all modes contributing to the experimental surfacedisplacements are identified separately and compared with the correspondingtheoretical dispersion curves obtained through equation (5.1) [22, 73, 128]. Thisapproach requires a large number of sensors. Furthermore, it is not always possibleto differentiate between dispersion curves of higher modes in the experimental data[127].

In the second strategy, Nazarian’s method [151] is used to derive a singleexperimental dispersion curve from the measured surface displacements, whichis compared with an effective theoretical dispersion curve that accounts for thedominance of higher modes. This approach does not exploit all informationprovided by the experimental data, but is more robust. The number of sensorsremains limited. Ganji et al. [74] have calculated the effective theoretical dispersioncurve by means of a numerical simulation of an actual SASW test and theapplication of Nazarian’s method to the simulated results. Gucunski and Woods[89] have inspected the wavenumber content of the soil’s response due to a verticalharmonic load at the soil’s surface in order to determine the dominant mode.

In the present work, a methodology similar to the approach of Zomorodian andHunaidi [233] is followed. The effective theoretical dispersion curve is calculated asCT

R (ω) = ω/kTR(ω). Here, kT

R(ω) is the horizontal wavenumber where the modulusof the Green’s function uG

zz(z′ = 0, kr, z = 0, ω) reaches its absolute maximum for

a fixed frequency ω.

Layer d Cs Cp β ρ[m] [m/s] [m/s] [−] [kg/m3]

1 10 400 800 0.030 18002 5 300 600 0.035 17003 10 400 800 0.030 18004 ∞ 500 1000 0.025 1800

Table 5.2: Benchmark soil profile 1.

The theoretical dispersion curve 109

Layer d Cs Cp β ρ[m] [m/s] [m/s] [−] [kg/m3]

1 2 180 300 0.010 18002 4 120 857 0.010 18003 8 180 1286 0.010 18004 ∞ 360 1323 0.010 1800

Table 5.3: Benchmark soil profile 2.

This approach gives acceptable results for soil profiles where a soft layer isoverlain by stiffer layers. As an example, the method is applied to two soilprofiles considered as benchmark problems in different papers on this subject [123,157, 233]. Both profiles consist of three homogeneous layers on a homogeneoushalfspace. The thickness d, shear wave velocity Cs, dilatational wave velocityCp, hysteretic material damping ratio β (for both shear and dilatational waves),and density ρ of the layers and halfspace are given in tables 5.2 and 5.3. Forboth profiles, the effective theoretical dispersion curve CT

R (ω) is derived fromthe wavenumber domain Green’s function uG

zz(z′ = 0, kr, z = 0, ω). A reference

solution is obtained from the numerical simulation of an SASW test with 10receivers, following the approach of Ganji et al. [74]. Figure 5.5 compares bothwith the theoretical dispersion curves of all modes obtained from equation (5.1).The dispersion curve of the fundamental Rayleigh wave differs from the referencesolution due to the dominance of higher modes. A satisfactory agreement betweenthe effective theoretical dispersion curve CT

R(ω) and the reference solution isobtained in both cases.

(a)0 25 50 75 100 125 150 175 200

0

150

300

450

600

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(b)0 25 50 75 100 125 150 175 200

0

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 5.5: Effective theoretical dispersion curve CTR (ω) obtained from the

wavenumber domain Green’s function uGzz(z

′ = 0, kr, z = 0, ω) (gray line)compared with a reference solution derived from the numerical simulation of anSASW test (black dots) and with the theoretical dispersion curves of all modes(thin black lines) for (a) soil profile 1 and (b) soil profile 2.

110 The SASW method

5.5 The inverse problem

The inversion procedure minimizes the distance between the theoretical dispersioncurve CT

R (ω) and the experimental dispersion curve CER(ω) shown in figure 5.4. The

objective function in the minimization scheme is defined as:

fobj =

√

√

√

√

1

N

N∑

n=1

(

CTR (ωn) − CE

R(ωn))2

(5.10)

where N = 20 logarithmically spaced frequencies ωn between 20 Hz and 166 Hz areused. A logarithmic spacing scheme is used to assign a higher weight to the lowfrequency range. In this way, the steep variation of the experimental dispersioncurve in the low frequency range is properly accounted for.

For the calculation of the theoretical dispersion curve, the soil is modelled asa horizontally layered linear elastic halfspace. Each layer is characterized by athickness d, a shear wave velocity Cs, a dilatational wave velocity Cp, a hystereticmaterial damping ratio β (for both shear and dilatational waves), and a density ρ.The parameters in the optimization are the layer thickness d and the shear wavevelocity Cs for each layer. The other characteristics are derived from the results ofearlier tests performed at the same site [110, 167]. For all layers, a Poisson’s ratioν = 0.33 (hence Cp = 2Cs) and a density ρ = 1800 kg/m3 is used. As the effectof the material damping ratio on the dispersion curve is negligible, an arbitrarilychosen value β = 0.03 is used for all layers.

The inversion is performed in different stages, increasing the number of layers.In each stage, an unconstrained non-linear least squares problem is solved in orderto minimize the objective function fobj. This problem is solved in MATLABby means of the Levenberg-Marquardt algorithm [144]. The initial values of theoptimization parameters in each stage are based on the final values of the previousstage. Table 5.4 gives these values for all stages, as well as the final value of theobjective function. The corresponding dispersion curves are shown in figure 5.6.

In stage 4, the soil is modelled with three layers on a halfspace. The inversionprocedure yields a thickness d of 0.57 m for the top layer, 1.44 m for the secondlayer, and 3.43 m for the third layer. The shear wave velocity Cs is 140 m/s in thetop layer, 154 m/s in the second layer, 220 m/s in the third layer, and 287 m/s inthe halfspace. For this profile, the objective function fobj equals 0.352 m/s and thetheoretical dispersion curve corresponds very well to the experimental dispersioncurve (figure 5.6d). The profile obtained in stage 4 of the inversion procedure istherefore considered as the solution of the inverse problem.

5.6 The foundation-soil transfer functions

In this section, the dynamic soil-structure interaction problem of section 4.4 isreconsidered. The homogeneous soil profile is replaced with the profile described

The foundation-soil transfer functions 111

Initial profile Final profileStage Layer d Cs d Cs fobj

[m] [m/s] [m] [m/s] [m/s]1 1 ∞ 150 ∞ 162 18.912

21 3.00 162 1.89 146

2.3192 ∞ 162 ∞ 228

31 1.89 146 0.84 142

0.7072 1.00 228 1.61 1623 ∞ 228 ∞ 240

4

1 0.84 142 0.57 140

0.3522 0.80 162 1.44 1543 0.80 162 3.43 2204 ∞ 240 ∞ 287

Table 5.4: Initial and final soil profile for all stages in the inversion procedure.

(a)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(b)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(c)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(d)0 25 50 75 100 125 150 175 200

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 5.6: Experimental (black dots) and theoretical (gray line) dispersion curvefor (a) a homogeneous halfspace, (b) a layer on a halfspace, (c) two layers on ahalfspace, and (d) three layers on a halfspace.

in table 5.5. The Poisson’s ratio ν (or equivalently, the ratio Cs/Cp) and thedensity ρ follow from earlier tests [110, 167] and correspond to the values usedin the inversion procedure in subsection 5.5 to determine the shear wave velocity

112 The SASW method

profile Cs. The damping ratio β is determined so that the predicted and themeasured displacements correspond well in the far field.

Layer d Cs Cp β ρ[m] [m/s] [m/s] [−] [kg/m3]

1 0.57 140 280 0.03 18002 1.44 154 308 0.03 18003 3.43 220 440 0.03 18004 ∞ 287 574 0.03 1800

Table 5.5: Soil profile used for the numerical vibration predictions.

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 5.7: Modulus of the measured (black line) and predicted (gray line)foundation-soil transfer function H(ω) for different receivers, located at (a) 4 m,(b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation. The modulusof the measured transfer function H(ω) is plotted as a solid line if the coherencefunction Γ(ω) between the hammer force and the free field response exceeds athreshold value Γmin = 0.95.

The dynamic foundation-soil interaction problem is solved in the same way asin section 4.4, using the same parameter values. Figure 5.7 compares the resultingfoundation-soil transfer function H(ω) with the experimental data introduced insection 1.4.

Conclusion 113

Up to 16 m from the foundation, the coherence Γ(ω) between the impact forceand the free field displacement is high between 20 and 150 Hz. At 32 m from thefoundation, the coherence function Γ(ω) decreases in the higher frequency range,indicating a lower signal-to-noise ratio. In the frequency range where the coherenceΓ(ω) is high, the measured results are reliable and can be compared with thepredicted results. In this frequency range, the correspondence of the measured andthe predicted transfer functions is acceptable up to 50 Hz. At higher frequencies,the numerical model leads to an overestimation of the free field displacement.

This overestimation appears to be due to an underestimation of the materialdamping ratio. The material damping ratio can not be solely responsible, however:a similar overestimation is observed at all source-receiver distances, while thematerial damping ratio especially affects the far field response. Increasing thematerial damping ratio in order to fit the predicted and measured response in thenear field would therefore lead to an underestimation of the measured response inthe far field.

The difference between the measured and predicted response might also be dueto inaccuracies in the identified soil profile: the identified soil profile is not theonly one that fits the experimental dispersion curve. In chapter 8, a probabilisticmethod is used to identify an ensemble of soil profiles that fit the experimentaldispersion curve. Each of these profiles is used to predict the foundation-soiltransfer functions. The variability of the transfer functions is considered to assessthe robustness of the SASW method for the prediction of ground vibrations.

Another possible explanation for the overestimation of the free field response isnon-linear behaviour of the soil under the foundation. This explanation is alsoput forward by Auersch [18], who observed a similar overestimation of measuredground vibrations in the higher frequency range.

An additional cause of the discrepancy between the predictions and themeasurements might be a (local) deviation from a horizontally layered soil profile.In the higher frequency range, the response of the foundation (and, consequently,the free field response) is to a large extent determined by the soil properties directlyunder the foundation. A local variation of the soil properties might therefore havean impact on the vibrations in the free field.

5.7 Conclusion

This chapter focuses on the SASW method. The SASW method allows todetermine the shear wave velocity of a layered soil as a function of depth. As a partof the present work, the SASW method is applied to the site in Lincent describedin section 1.4. The identified soil profile is used to simulate the foundation-soiltransfer functions measured on this site. The simulated results are comparedwith experimental data in order to validate the identified soil profile. Whilean acceptable correspondence of predicted and measured data is observed in the

114 The SASW method

frequency range between 20 Hz and 50 Hz, an important discrepancy emerges inthe higher frequency range.

It is no trivial task to point out the errors in the identified soil profile that give riseto the deviations from the measurement data. Possible causes include the inversionprocedure giving erroneous results, an underestimation of the material dampingratio, non-linear behaviour of the soil under the foundation, and variations ofthe soil properties in the horizontal direction. The robustness of the inversionprocedure is investigated in chapter 8 where the inversion problem is addressedin a probabilistic way. In the preceding chapters 6 and 7, the stochastic finiteelement method is introduced and used to study wave propagation in a soil withuncertain properties.

Chapter 6

Stochastic mechanics

6.1 Introduction

Since high performance computers have become available, it is possible to solvecomplex mechanical problems with a precision that reaches far beyond theavailability of information on both the system properties and the excitation. As aconsequence, the field of stochastic mechanics has received considerable attention.This has led to the development of some practical tools to account for uncertaintyin mechanical problems. The review paper by Schueller [190] gives an overview ofthe 1997 state of the art in this field. More recent advances are addressed in aspecial issue of Computer Methods in Applied Mechanics and Engineering [191].

Classically, a probabilistic approach is followed and the uncertain properties arecharacterized by mean values, variances, covariances, probability distributions, andjoint probability distributions [119]. More recently, non-probabilistic approacheshave been developed such as interval and fuzzy analysis, where uncertainparameters are represented by intervals or fuzzy sets [149]. In this work, theuncertainty is quantified in a probabilistic way. This approach allows to computethe standard deviation and the confidence region of the response of a soil to a givenexcitation, which are suitable measures for the response variability. Moreover,the probabilistic approach provides a rigorous theoretical framework to modelrandom processes, which are used in this work to represent the spatial variationof soil properties. An apparent drawback of the probabilistic approach is theneed to formulate a Probability Density Function (PDF) instead of an intervalfor the uncertain parameters, which may be complicated. The maximum entropyprinciple can be used to select an appropriate PDF [107, 198]. Given the availableinformation on a random parameter (e.g. the mean value and the standarddeviation), this principle allows to derive the PDF that maximizes the uncertainty.

115

116 Stochastic mechanics

Mathematically, a linear mechanical problem subjected to uncertainty can bemodelled by the following partial differential equation:

Lu = f (6.1)

where L is a linear partial differential operator that depends on the systemproperties, f is the excitation, and u is the response. If either the operator L,the excitation f , or both are uncertain then this uncertainty has an impact onthe response u through equation (6.1). The case where a deterministic systemrepresented by a deterministic operator L is subjected to an uncertain excitation fhas been extensively studied [153]. By virtue of the linear relationship between theuncertain excitation and the response, these so-called random vibration problemscan often be solved analytically. The case where the operator L is not deterministicis more complicated. The solution strategies for random vibration problems arenot applicable in this case due to the non-linear relationship between the uncertainparameters and the response.

The mathematical-mechanical modelling process that leads to the differentialequation (6.1) or an equivalent system of algebraic equations introduces twotypes of uncertainties: data uncertainties and model uncertainties. Datauncertainties are related to the parameters of the model such as the geometryand the elasticity tensor. These uncertainties can be modelled using the classicalparametric probabilistic approach where they are represented by random variablesor processes. This approach does not allow to account for model uncertainties,however. Therefore, a non-parametric approach has recently been developed bySoize [200] using random matrix theory [199, 201] and the maximum entropyprinciple [107, 198]. An inherent drawback of the non-parametric approach isthe vague physical meaning of the uncertainty in the model that complicates theinterpretation of the results. Arnst et al. [12] apply the non-parametric approachto a dynamic soil-structure interaction problem using random matrices to modelthe modal stiffness, damping, and mass of the structure. In the present work,the classical parametric approach is followed and model uncertainties are notaccounted for.

In 1991, Ghanem and Spanos [78] introduced the Stochastic Finite ElementMethod (SFEM), which has become one of the most widely used methods tomodel uncertainty in computational mechanics. In the SFEM, the uncertainparameters in the system or the excitation are expressed in terms of a limitednumber of underlying random variables. These variables determine the responseof the system, which can therefore be expressed as a response surface in termsof the random variables. The response surface is either randomly sampled orapproximated by a polynomial in order to estimate the variability of the response.

In this chapter, the SFEM is reviewed. Elements of the SFEM are used in thefollowing chapters to study wave propagation in a soil with uncertain properties.The chapter is organized as follows. Section 6.2 introduces the Kolmogorovprobability theory which serves as a mathematically rigorous framework todescribe random variables and random processes. Sections 6.3 and 6.4 focus

Kolmogorov’s probability theory 117

on methods to simulate random vectors and random processes with prescribedmarginal probability distributions and a prescribed correlation structure. Thesemethods are used in the SFEM, which is discussed in section 6.5.

6.2 Kolmogorov’s probability theory

Since Lebesgue introduced his measure and integration theory in the beginningof the 20th century, several mathematicians have attempted to formulate aprobability theory based on the same viewpoints. Kolmogorov was the first topresent a consistent exposition of this theory in 1933 [119]. Recently, Dooboutlined the development of Kolmogorov’s probability theory [55]. He used anotation that differs slightly from Kolmogorov’s original notation but agreeswith the current practice in the literature. In this section, some elements ofKolmogorov’s probability theory are briefly outlined, following Doob’s notation.The purpose of this section is to introduce some probability theoretical conceptsoften relied upon in the literature on probabilistic mechanics.

6.2.1 The probability space

The context of problems involving uncertainty is the probability space (Ω, S, P ),consisting of the sample space Ω, the event space S, and the probability measure P .

The sample space Ω is a set that contains the elementary events θ. Theelementary events determine the state of the random problem: each elementaryevent corresponds to a single possible state. As the number of possible states of aphysical problem can be infinite, the sample space Ω can be an infinite set. Withineach state, the random variables under consideration take a fixed, deterministicvalue.

The event space S represents a collection of subsets S ⊂ Ω called events. Anexample is the event Si that a given random variable takes a value in a certaininterval. This event Si contains all elements θ ∈ Ω corresponding to states of therandom problem for which the random variable under consideration takes a valuein that interval. Technically, the event space S is a σ-algebra over the set Ω, whichmeans that it contains (1) the set Ω itself, (2) the complement Ω \ S of each setS ∈ S, and (3) the union

⋃

k Sk of each sequence of sets Sk ∈ S.The measure P assigns a probability P (S) between 0 and 1 to each event S ∈ S.

The elementary event θ that corresponds to the actual (but unknown) state of therandom problem belongs to the subset S with a probability P (S). The probabilitiesof the improper subsets ∅ and Ω are P (∅) = 0 and P (Ω) = 1. If the subsets S1

and S2 have no elements in common, then P (S1 ∪S2) = P (S1)+P (S2). An eventS is said to happen almost surely if it has a probability P (S) = 1.

118 Stochastic mechanics

6.2.2 Random variables

Definition

In order to define an n-variate random vector V, the random vector’s state space(Rn,BRn) is first constructed. R

n is the set of n-dimensional real vectors and BRn

is the Borel σ-algebra over the set Rn. The Borel σ-algebra over R

n is the minimalσ-algebra containing all open subsets of R

n or, equivalently, all closed subsets, as aσ-algebra also contains the complements of its elements. Next, the random vectorV is defined as a function from the sample space Ω to the set of real vectors R

n:

θ 7→ V(θ) : Ω → Rn (6.2)

This function assigns a value v = V(θ) to each elementary event θ and a subsetB ∈ BRn to each event S ∈ S, as illustrated in figure 6.1. The function V(θ)

Ω

S

θ

Rn

v

B

Figure 6.1: Kolmogorov’s representation of a random vector v : Ω → Rn.

must be Borel measurable, which means that the inverse image S = V−1(B) ofeach subset B ∈ BRn must belong to the σ-algebra S. In this way, the probabilityfunction PV(B) of the random vector V(θ) can be defined for each subset B ∈ BR

as:

PV(B) = P (V−1(B)) (6.3)

The function PV(B) gives the probability that the value v of the random vectorV(θ) belongs to the subset B.

Probability distribution

The Cumulative Distribution Function (CDF) FV of the random vector V(θ) isdefined as:

FV(v) = PV(v′ ∈ Rn : v′i < vi for all i = 1, . . . , n) (6.4)

As the set v′ ∈ Rn : v′i < vi for all i = 1, . . . , n is a subset of the Borel σ-

algebra BRn , the requirement that V(θ) is Borel measurable guarantees that theprobability on the right hand side of equation (6.4) is defined. It follows from

Kolmogorov’s probability theory 119

equation (6.4) that the CDF FV(v) increases monotonically from 0 at −∞ to 1 at+∞.

If the CDF FV(v) is differentiable in Rn, the PDF pV(v) of the random vector

V(θ) can be defined as:

pV(v) =∂n

∂v1 · · · ∂vnFV(v) (6.5)

The PDF pV(v) is therefore a non-negative function that integrates to one. TheCDF FV(v) and the PDF pV(v) of the n-variate random vector V(θ) are oftenreferred to as the joint CDF and the joint PDF of the univariate random variablesVi(θ).

In probability theory, the notion of independent random variables is essential.The random variables V1(θ), . . . , Vn(θ) are mutually independent if and only iftheir joint PDF pV(v) can be expressed as:

pV(v) = pV1(v1) × · · · × pVn(vn) (6.6)

Within the framework of Kolmogorov’s probability theory, complex randomvariables cannot be defined in a direct way. An n-variate complex random vectorVc(θ) must be interpreted as a function of a 2n-variate real-valued random variableVi(θ),Vr(θ)T defined as:

Vc(θ) = Vr(θ) + iVi(θ) (6.7)

It is clear that the probability distribution of a complex random vector Vc(θ) isgiven by a 2n-dimensional CDF FVc(vr,vi) or PDF pVc(vr,vi).

Some examples of frequently used probability distributions are given insubsection 6.2.4.

Mathematical expectation

The mathematical expectation of a function f(V(θ)) of a random vector V(θ) isgiven by the following Lebesgue integral:

Ef(V(θ)) =

∫

Ω

f(V(θ))P (dθ) (6.8)

Using equation (6.3), this becomes:

Ef(V(θ)) =

∫

Rn

f(v)PV(dv) (6.9)

According to equations (6.4) and (6.5), the probability PV(dv) of the infinitesimalset dv is equal to FV(v + dv) − FV(v) = pV(v)dv, so:

Ef(V(θ)) =

∫

Rn

f(v)pV(v) dv (6.10)

120 Stochastic mechanics

The mathematical expectation operator is used to equip the vector space ofrandom variables V (θ) with an inner product:

〈V1(θ), V2(θ)〉 = EV1(θ)V∗2 (θ) (6.11)

This inner product allows to define distance and orthogonality in the vector spaceof random variables.

Statistical properties

The p-th order statistical moment mV p of a univariate random variable V (θ) isdefined as:

mV p = EV p(θ) (6.12)

For an n-variate random vector V(θ), the order p becomes an n-dimensional multi-index and the statistical moment mVp of order p is defined as:

mVp = EV p11 (θ) × · · · × V pnn (θ) (6.13)

The mean value mV and the variance σ2V of the univariate random variable V (θ)

are defined as:

mV = EV (θ) (6.14)

σ2V = E|V (θ) −mV |2 = E|V (θ)|2 − |mV |2 (6.15)

where σV is the standard deviation. The standard deviation σV represents theroot mean square distance between the random variable V (θ) and its mean valuemV , where the distance is measured along the real line for real random variablesand in the complex plane for complex random variables. For positive real-valuedrandom variables, the variability can be characterized by means of the coefficientof variation, defined as the ratio σV /mV .

The mean value mV of a random vector V(θ) is defined in an analogous way asfor a random variable:

mV = EV(θ) (6.16)

The correlation matrix RV and the covariance matrix CV of the random vectorV(θ) are defined as:

RV = EV(θ)VH(θ) (6.17)

CV = EV(θ)VH(θ) − EV(θ)EV(θ)H (6.18)

where VH(θ) denotes the Hermitian or conjugate transpose of V(θ). Thecovariance matrix CV can be normalized with respect to the standard deviations

Kolmogorov’s probability theory 121

σViof the random variables Vi(θ):

[rV]ij =[CV]ijσVi

σVj

(6.19)

The element [rV]ij of the matrix rV is called the correlation coefficient of therandom variables Vi(θ) and Vj(θ). The correlation coefficient [rV]ij takes a valuebetween −1 and 1. If the correlation coefficient [rV]ij is zero, the random variablesVi(θ) and Vj(θ) are uncorrelated. Whereas independent variables are alwaysuncorrelated, the opposite is not generally true.

The matrices RV, CV, and rV are symmetric and positive semidefinite. Thesymmetry directly follows from the definitions given in equations (6.17–6.19).The positive semidefiniteness can be shown using an auxiliary univariate randomvariable U(θ) = aHV(θ), where a is an arbitrary vector. The variance σ2

U of thisvariable is given by:

σ2U = E U(θ)U∗(θ) − |mU |2 (6.20)

= E

aHV(θ)VH(θ)a

− |mU |2 (6.21)

= aHRVa − |mU |2 (6.22)

Since σ2U and |mU |2 are by definition non-negative, it follows from equation (6.22)

that:

aHRVa ≥ 0 (6.23)

As equation (6.23) is valid for any vector a, the correlation matrix RV is a positivesemidefinite matrix. The positive semidefiniteness of the matrices CV and rV canbe shown in a similar way.

6.2.3 Random processes

Definition

An n-variate random process W on a d-dimensional domain D ⊂ Rd is defined as

a Borel measurable function W(x, θ) from the probability space (Ω, S, P ) to thestate space (H,BH):

θ 7→ W(x, θ) : Ω → H (6.24)

where H is a vector space of functions f : D → Rn and BH is the Borel σ-algebra

over H. The probability function PW(B) of the random process is defined for eachsubset B ∈ BH as:

PW(B) = P (W−1(B)) (6.25)

122 Stochastic mechanics

Probability distribution

It is impossible to define a CDF or PDF for a random process in a similar wayas for a random variable. In order to characterize the probability distribution ofthe random process W(x, θ), it can be regarded as a family of random variablesWx(θ) : x ∈ D [161]. The domain D is referred to as the index set. Foreach finite subset x1, . . . ,xp of the index set D, the corresponding np-variaterandom vector Wx1(θ), . . . ,Wxp

(θ)T can be characterized by an np-dimensionalCDF or PDF. If the subset of D consists of a single index x, these functionsare called the marginal CDF and PDF of the random process. The completestochastic characterization of a random process requires the definition of allits finite-dimensional CDFs or PDFs, which is unachievable in practice. As aresult, the stochastic characterization of a physical random process is by definitionincomplete.

The random process W(x, θ) is stationary of order p if the random vectorsWx1(θ), . . . ,Wxp

(θ)T and Wx1+∆x(θ), . . . ,Wxp+∆x(θ)T are identically dis-tributed for all x1, . . . ,xp,∆x ∈ D. In order for Wx1+∆x(θ), . . . ,Wxp+∆x(θ)T

to be defined, the index set D must be linear, which means that for all x,∆x ∈ D,x+∆x must also belong to D. A process is weakly stationary if it is stationary oforder 1 and 2. It is strictly stationary if it is stationary of order p for all integers p.

Mathematical expectation

The mathematical expectation of a function f(W(x, θ)) of a random processW(x, θ) is defined in a similar way as for a random vector:

Ef(W(x, θ)) =

∫

Rn

f(w)pWx(w) dw (6.26)

where pWx(w) denotes the marginal PDF of the random process W(x, θ) at the

index x.

Statistical properties

The mean value mW (x) and the standard deviation σW (x) of the univariaterandom process W (x, θ) are defined as:

mW (x) = EW (x, θ) (6.27)

σ2W (x) = E|W (x, θ) −mW (x)|2 = E|W (x, θ)|2 − |mW (x)|2 (6.28)

For weakly stationary random processes, the mean value mW (x) and the standarddeviation σW (x) are constant with respect to x.

Kolmogorov’s probability theory 123

The correlation function RW (x1,x2) and the covariance function CW (x1,x2) ofthe univariate random process W (x, θ) are defined as:

RW (x1,x2) = EW (x1, θ)W∗(x2, θ) (6.29)

CW (x1,x2) = EW (x1, θ)W∗(x2, θ) − EW (x1, θ)EW (x2, θ)∗ (6.30)

For weakly stationary random processes, these functions only depend on therelative distance x = x2 − x1 and can be denoted by RW (x) and CW (x).

In a similar way as for a random vector V(θ), the correlation and covariancefunctions RW (x1,x2) and CW (x1,x2) of a random process are symmetric positivedefinite functions.

While the marginal PDF pWx(w) of a random process W(x, θ) determines its

behaviour at a fixed position x, the covariance function CW (x) (or, equivalently,the correlation function RW (x)) determines how the process varies with respectto x. In particular, the covariance function determines the scale of variation andthe smoothness of the random process.

For a univariate weakly stationary random process defined on a one-dimensionaldomain, the scale of variation is given by the correlation length lc. Following Soize[201], the following definition is used:

lc =1

σ2W

∫ L

0

CW (x) dx (6.31)

Random processes with a small correlation length lc vary rapidly with x, whilerandom processes with a large correlation length lc vary slowly with x.

The smoothness of a random process is related to its continuity and differentia-bility. Following Banerjee and Gelfand [20], three notions of process smoothnessare considered: almost sure continuity, mean square continuity, and mean squaredifferentiability.

A process W (x, θ) is almost surely continuous at x0 if W (x, θ) tends to W (x0, θ)almost surely (i.e. with probability 1) when x tends to x0. If the processW (x, θ) isalmost surely continuous at every x0 ∈ D, it is said to have continuous realizations[20].

A process W (x, θ) is mean square continuous or L2 continuous at x0 if:

limx→x0

E

|W (x) −W (x0)|2

= 0 (6.32)

A process W (x, θ) is mean square differentiable if there exists a process W ′(x, θ)such that:

lim∆x→0

E

∣

∣

∣

∣

W (x+ ∆x, θ) −W (x, θ)

∆x−W ′(x, θ)

∣

∣

∣

∣

2

= 0 (6.33)

If the process W ′(x, θ) is also mean square differentiable, the process W (x, θ) istwice mean square differentiable. Higher order mean square differentiability isdefined analogously.

124 Stochastic mechanics

The smoothness properties of a random processW (x, θ) depend on its covariancefunction CW (x1,x2). The exact conditions for the covariance function are givenby Stein [205] and Kent [116] and summarized by Banerjee and Gelfand [20].

6.2.4 Probability distributions

This subsection gives an overview of the probability distributions used in thepresent work. For each distribution, expressions are given for the mean value andthe standard deviation.

The Gaussian probability distribution

The univariate Gaussian (or normal) PDF pG(v|m,σ) with a mean value m and astandard deviation σ is given by:

pG(v|m,σ) =1

σ√

2πe−

(v−m)2

2σ2 =1

σpG

(

v −m

σ

)

(6.34)

Herein, the function pG(ξ) is the standard Gaussian PDF, with a zero mean valueand a unit standard deviation:

pG(ξ) =1√2πe−

ξ2

2 (6.35)

The n-variate Gaussian PDF pG(v|m,C) with mean m and correlation C isdefined as:

pG(v|m,C) =1

(2π)n2

√detC

e−12 (v−m)TC

−1(v−m) (6.36)

In the bivariate case, the covariance matrix C has the following structure:

C =

[

σ21 rσ1σ2

rσ1σ2 σ22

]

(6.37)

where r is the correlation coefficient. In the case where v1(θ) and v2(θ) arecorrelated standard Gaussian variables, their PDF is denoted in the followingas pG(v1, v2|r).

The Gaussian probability distribution has some interesting properties. First,if the variables vi(θ) are mutually uncorrelated, the covariance matrix C has adiagonal structure C = diagσ2

i and equation (6.36) can be reformulated as:

pG(v|m,C) =

n∏

i=1

(

1

σi√

2πe−

(vi−mi)2

2σ2i

)

(6.38)

This equation shows that the joint PDF of a set of uncorrelated Gaussian variablesvi(θ) can be written as a product of one-dimensional PDFs. For Gaussian variables,uncorrelatedness therefore implies independence.

Kolmogorov’s probability theory 125

Second, an affine transformation V′(θ) = AV(θ) +B of a Gaussian vector V(θ)is again a Gaussian vector V′(θ) with a mean value mV′ = AmV + B and acovariance matrix CV′ = ACVAT [198].

Third, according to the central limit theorem, the PDF of a random variableobtained as the sum of a large number of independent random variables (witharbitrary probability distributions) tends to the Gaussian PDF [65].

The Gaussian probability distribution is frequently adopted to characterize theuncertainty in mechanical models as it leads to tractable numerical formulationsdue to the properties discussed above. Moreover, this choice is often motivatedby the central limit theorem, as the uncertainties are assumed to stem from alarge number of (independent) sources. The Gaussian probability distribution isnot always physically realistic, however. If the stiffness of a system is modelled asa Gaussian variable, the variance of the static response of the system is infinite,which is impossible [186, 198]. This is due to the finite probability that a Gaussianvariable takes a value near zero. Furthermore, a Gaussian stiffness takes a negativevalue with finite probability, which is also unrealistic.

The lognormal probability distribution

A random variable V (θ) is lognormally distributed if its logarithm ln(V (θ)) is aGaussian variable. The parameters in the lognormal distribution are the meanvalue M and the standard deviation S of the underlying Gaussian variable. Thelognormal PDF pL(v|M,S) is equal to:

pL(v|M,S) =1

S√

2πe−

(ln(v)−M)2

2S2 (6.39)

The corresponding mean value mV and standard deviation σV are given by:

mV = eM+ S2

2 (6.40)

σ2V = eS

2+2M (eS2 − 1) (6.41)

The lognormal probability distribution is suitable to model properties that arealmost surely positive, such as the stiffness of a mechanical system.

The uniform probability distribution

A random variable V (θ) follows the uniform probability distribution if it almostsurely belongs to a given interval [a, b] and any value in the interval [a, b] is takenwith equal probability. The uniform PDF pU(v|a, b) is defined as:

pU(v|a, b) =

1b−a if a ≤ v ≤ b

0 otherwise(6.42)

126 Stochastic mechanics

The mean value mV and the standard deviation σV are given by:

mV =1

2(a+ b) (6.43)

σ2V =

1

12(b− a)2 (6.44)

The uniform probability distribution allows to control the probabilistic informationintroduced in a mechanical model in a conceptually very simple way, imposing hardbounds on the uncertain parameters.

The beta probability distribution

A beta distributed random variable V (θ) almost surely takes a value in the interval[0, 1], but not all values in this interval are equally probable. The PDF pB(v)depends on two parameters α and β and is defined as:

pB(v) =

1B(α,β)v

α−1(1 − v)β−1 if 0 ≤ v ≤ 1

0 otherwise(6.45)

The beta function B(α, β) serves as a normalization constant such that the PDFpB(v) integrates to one.

The mean value mV and the standard deviation σV are given by:

mV =α

α+ β(6.46)

σ2V =

αβ

(α+ β + 1)(α+ β)2(6.47)

Like the uniform PDF, the beta PDF can be used to model uncertain parametersthat take a value in a bounded interval almost surely (such as the Poisson’s ratio).

6.2.5 Covariance functions

In this subsection, three frequently used covariance functions for weakly stationaryprocesses are introduced. For each covariance function, the correlation length iscalculated and the smoothness of the process is discussed. The scope of thissubsection is limited to one-dimensional processes.

The Matern covariance function

The Matern covariance function is one of the most important covariance functionsused to model physical phenomena [92, 95, 197]. The Matern covariance function

Kolmogorov’s probability theory 127

CM(x) of the weakly stationary random process W (x, θ) with standard deviationσW is defined as:

CM(x) = σ2W

21−n

Γ(n)(αx)nKn(αx) (6.48)

where Kn is a modified Bessel function of the second kind and α and n are strictlypositive parameters that determine the scale of variation and the smoothness ofthe random process W (x, θ). Due to the presence of these two parameters, theMatern covariance function offers much flexibility in stochastic modelling, whichexplains its popularity.

The scale of variation of the processW (x, θ) is reflected by the correlation lengthlc:

lc =2n−1

√πΓ(

n+ 12

)

α(6.49)

The continuity and differentiability of the random process W (x, θ) depends onthe parameter n. According to Kent [116], the process W (x, θ) has discontinuousrealizations if 0 < n ≤ 0.5 and continuous realizations if n > 0.5. As stated byBanerjee and Gelfand [20] and Simak [197], the process W (x, θ) is always meansquare continuous, but it is only k times mean square differentiable if n > k.

The exponential covariance function

The exponential covariance function CE(x) of the weakly stationary randomprocess W (x, θ) with standard deviation σW is defined as:

CE(x) = σ2W e

| xlc| (6.50)

where lc is the correlation length. The exponential covariance function CE(x)is in fact a special case of the Matern covariance function CM, for which thesmoothness parameter n equals 0.5. A random processW (x, θ) with an exponentialcovariance function CE(x) is therefore mean square continuous but not meansquare differentiable, and has discontinuous realizations.

The Gaussian covariance function

The Gaussian covariance function CG(x) of the weakly stationary random processW (x, θ) with standard deviation σW is defined as:

CG(x) = σ2W e

α2x2

(6.51)

The corresponding correlation length lc is given by:

lc =

√π

2α(6.52)

128 Stochastic mechanics

The Gaussian covariance function is a special case of the Matern covariancefunction, for which the smoothness parameter n tends to infinity. A randomprocess W (x, θ) with a Gaussian covariance function CE(x) is consequently verysmooth: it is infinitely many times mean square differentiable and has continuousrealizations.

6.3 Random vector modelling

A random vector V(θ) is completely characterized by its joint PDF pV(v). If thevector V(θ) represents a set of physical quantities, it is often difficult to determinethis joint PDF pV(v). It is therefore common in the literature on probabilisticmechanics to characterize the random vector V(θ) in an incomplete way, usingonly a set of marginal PDFs pVk

(vk) and a correlation matrix RV or a covariancematrix CV.

For many methods in the field of probabilistic mechanics, it is necessary that theuncertainty on the system properties is expressed in terms of a set of independentGaussian variables. This section describes how a random vector V(θ) withprescribed marginal PDFs pVk

(vk) and a prescribed correlation structure RV canbe modelled as a function of a set of independent Gaussian variables ξk(θ). Theresulting model can also be used to simulate the vector V(θ), using a (pseudo-)random number generator to generate realizations of the independent Gaussianvariables ξk(θ).

Subsection 6.3.1 addresses the decorrelation of a set of correlated randomvariables Vk(θ): the random variables Vk(θ) are expressed as a linear combinationof a set of uncorrelated random variables ξk(θ). The decorrelation procedure onlyyields a useful model of the random vector V(θ) if its components Vk(θ) follow theGaussian distribution.

In subsection 6.3.2, the Nataf transformation is introduced. The Nataftransformation allows to express a set of non-Gaussian variables Vk(θ) withprescribed marginal PDFs pVk

(vk) and a prescribed correlation matrix RV asa non-linear function of a set of correlated Gaussian variables ηk(θ). It is shownhow the correlation matrix Rη of the Gaussian variables ηk(θ) can be determinedso that the non-linear function yields non-Gaussian variables Vk(θ) following thetarget correlation structure RV. In a subsequent step, the decorrelation procedurediscussed in subsection 6.3.1 can be applied to the correlated Gaussian variablesηk(θ) in order to obtain a usable model of the non-Gaussian random vector V(θ).

6.3.1 The decorrelation of random variables

The aim of a decorrelation procedure is to express a set of n correlated randomvariables Vk(θ) as a linear combination of n uncorrelated random variables ξk(θ).The random variables Vk(θ) and ξk(θ) are collected in the random vectors V(θ)and ξ(θ), respectively.

Random vector modelling 129

The decorrelation of random variables is based on the following eigenvalueproblem:

RVΦ = λΦ (6.53)

This problem leads to the eigenvalues Λ = diagλk and eigenvectors Φ =Φ1 · · ·Φn of the correlation matrix RV. The correlation matrix RV allowsthe following eigen decomposition:

RV = ΦΛΦT (6.54)

The eigenvectors Φk are mutually orthogonal as the correlation matrix RV issymmetric. The eigenvectors are also normalized, so that:

ΦTΦ = I (6.55)

The eigenvectors Φk form an orthonormal basis of the vector space Rn. The

random vector V(θ) can therefore be expressed as a linear combination of theeigenvectors Φk with random coefficients ξk(θ):

V(θ) = ΦΛ12 ξ(θ) (6.56)

where it is understood that Λ12 denotes the matrix diagλ

12

k . An expression for thecoefficients ξ(θ) is obtained from equation (6.56) by means of a left multiplicationwith ΦT:

ξ(θ) = Λ− 12 ΦTV(θ) (6.57)

The mean value of the coefficients ξk(θ) is found as:

Eξ(θ) = Λ− 12 ΦTmV (6.58)

where mV is the mean value of the vector V(θ). Using equations (6.17), (6.54),(6.55), and (6.57), it can be shown that the random coefficients ξk(θ) are mutuallyuncorrelated:

Eξ(θ)ξT(θ) = I (6.59)

For Gaussian variables Vk(θ), it follows from equation (6.57) that the uncorrelatedvariables ξk(θ) are also Gaussian. In such case, the variables ξk(θ) are mutuallyindependent. In a Monte Carlo simulation, realizations of independent Gaussianvariables ξk(θ) are easily obtained by means of a (pseudo-)random numbergenerator. These realizations can subsequently be introduced in equation (6.56)to obtain realizations of the correlated Gaussian variables Vk(θ). For randomvariables Vk(θ) with a non-Gaussian distribution, the uncorrelated randomvariables ξk(θ) are non-Gaussian and mutually dependent. As a result, they cannot be directly simulated by means of a (pseudo-)random number generator.

130 Stochastic mechanics

6.3.2 The Nataf transformation

The Nataf transformation [130] is used to model a set of n non-Gaussian randomvariables Vk(θ) with prescribed marginal probability distributions pVk

(vk) and aprescribed correlation matrix RV. The non-Gaussian variables Vk(θ) are expressedas a function of n correlated standard Gaussian random variables ηk(θ) with acorrelation matrix Rη:

Vk(θ) = gk(

ηk(θ))

= F−1Vk

(

FG

(

ηk(θ))

)

(6.60)

The definition of the transformation gk in terms of the marginal cumulative densityfunction FVk

(vk) of the non-Gaussian variable Vk(θ) and the standard Gaussiancumulative density function FG(ηk) ensures that the non-Gaussian variable Vk(θ)follows the target marginal probability distribution pVk

(vk). The correlationmatrix Rη of the underlying Gaussian variables ηk(θ) is chosen so that the non-Gaussian variables Vk(θ) are distributed according to the target correlation matrixRV. Hence, the correlation matrix Rη must satisfy the following equation:

[RV]ij = EVi(θ)Vj(θ) (6.61)

= E

gi(

ηi(θ))

gj(

ηj(θ))

(6.62)

=

∫∫

R2

gi(η1)gj(η2)pG

(

η1, η2| [Rη]ij

)

dη1dη2 (6.63)

where pG(η1, η2| [Rη]ij) denotes the joint PDF of a bivariate standard Gaussian

vector with correlation coefficient [Rη]ij . Equation (6.63) must be solvediteratively for fixed values of the indices i and j to obtain all elements of thecorrelation matrix Rη. The solution Rη only exists if the following condition ismet [83]:

[RV]ij ≥ E

gi(

η(θ))

gj(

−η(θ))

(6.64)

where η(θ) is a standard Gaussian variable. If this condition is met and the matrixRη is positive semidefinite, it is the correlation matrix of the underlying Gaussianvector η(θ).

In each iteration step performed to solve equation (6.63), the double integral onthe right hand side of the equation has to be evaluated. To this end, the Gaussianvariables ηi(θ) and ηj(θ) are first decorrelated by means of the procedure outlinedin subsection 6.3.1. This leads to:

ηi(θ)ηj(θ)

=1√2

√

1 + [Rη]ij

√

1 − [Rη]ij

√

1 + [Rη]ij −√

1 − [Rη]ij

ξi(θ)ξj(θ)

(6.65)

Random process modelling 131

where ξi(θ) and ξj(θ) are mutually independent standard Gaussian variables.Using equation (6.65), equation (6.62) is rewritten as:

[RV]ij = E

g′i(

ξi(θ), ξj(θ))

g′j(

ξi(θ), ξj(θ))

(6.66)

=

∫∫

R2

g′i(ξ1, ξ2)g′j(ξ1, ξ2)pG(ξ1)pG(ξ2) dξ1dξ2 (6.67)

where pG(ξ) denotes the standard Gaussian PDF. The integrals in equation (6.67)are evaluated using Gauss-Hermite quadrature [1]:

[RV]ij =

N∑

m=1

N∑

n=1

wGmw

Gn g

′i(ξ

Gm, ξ

Gn )g′j(ξ

Gm, ξ

Gn ) (6.68)

where ξGn and wGn are the Gaussian points and weights, and N is the number of

Gaussian points. The Gaussian points ξGn correspond to the roots of a Hermitepolynomial ΦN (ξ) of order N , and the Gaussian weights are given by:

wGn =

N !

N2Φ2N−1(ξ

Gn )

(6.69)

The Hermite polynomials Φp(ξ) are recursively defined as:

Φ0(ξ) = 1 (6.70)

Φ1(ξ) = ξ (6.71)

Φp+1(ξ) = ξΦp(ξ) − pΦp−1(ξ) (6.72)

6.4 Random process modelling

Random processes can be modelled in a similar way as random vectors, using aparameterized (or discretized) expression where the parameters are independentGaussian variables. This section focuses on some methods to model weaklystationary random processes W (x, θ) characterized by a marginal PDF pW (w)and a covariance function CW (x).

Modelling a Gaussian process is less complicated than a non-Gaussian process.Numerous methods for the discretization and simulation of Gaussian processeshave been presented in the literature. Three frequently used methods inthe field of stochastic mechanics are the Karhunen-Loeve decomposition, thespectral representation model, and the AutoRegressive Moving Average (ARMA)algorithm.

The Karhunen-Loeve decomposition of a Gaussian process [78] is similar tothe decorrelation of a Gaussian vector discussed in subsection 6.3.1. Thedecomposition consists of a linear combination of deterministic functions with

132 Stochastic mechanics

random coefficients. The deterministic functions are derived from the correlationfunction of the random process and the random coefficients are independentGaussian variables. The Karhunen-Loeve decomposition is discussed in more detailin subsection 6.4.1.

The spectral representation of a Gaussian process is a superposition of harmonicfunctions with random phases, random phases and amplitudes, or random phases,amplitudes, and frequencies [84]. The mean square amplitudes of the harmonicsdepend on the covariance function of the target random process.

The ARMA algorithm generates the current value of a Gaussian process asthe sum of a linear combination of its previous values and a linear combinationof uncorrelated Gaussian variables [181, 202]. Under certain conditions [181], aprocess modelled by the ARMA algorithm can be regarded as the response ofa linear system to a Gaussian white noise excitation. The parameters in theARMA algorithm can be chosen so that the covariance function of the realizationsapproaches the target covariance function. The computer memory requirementsfor the ARMA algorithm to simulate a long-duration process are limited, whichwas considered a major advantage twenty years ago. Today, much more computermemory is available and the ARMA algorithm has lost some of its importance inthe field of stochastic mechanics.

Due to the conceptual simplicity of the simulation models for Gaussian processes,Gaussian processes are often used to model the random properties of a mechanicalsystem. However, as indicated in subsection 6.2.4, this practice is in some casesphysically unrealistic. As a result, the development of simulation models for non-Gaussian processes recently received much attention.

Grigoriu [87] distinguishes three categories of simulation models for non-Gaussian processes: (1) translation processes, (2) conditional Gaussian processes,and (3) diffusion and filtered Poisson processes.

Translation processes are based on the non-linear transformation of an un-derlying Gaussian process [83]. This approach is very similar to the Nataftransformation of random vectors. First, the relation between the underlyingGaussian process and the target non-Gaussian process is formulated. Second,the covariance function of the underlying Gaussian process is determined so thatthe transformation leads to a non-Gaussian process with the target covariancefunction. Third, the underlying Gaussian process is modelled using any ofthe methods discussed above. The translation process method allows to modelprocesses with any marginal PDF and a broad range of covariance functions. Thismethod is addressed in subsection 6.4.2

Conditional Gaussian processes are non-Gaussian processes obtained withrandomized models for Gaussian processes [85]. As an example, the spectralrepresentation method can be randomized as follows: first, the covariance functionof the non-Gaussian process is defined as a function of a multidimensional randomvariable V(θ). Next, a realization of the random variable V(θ) is generated andused to obtain a realization of the covariance function. Finally, this covariancefunction is used to generate a realization of the random process by means of the

Random process modelling 133

classical spectral representation method. The PDF of the random variable V(θ)can be determined so that the realizations have the same covariance function anda similar marginal PDF as the target process.

Diffusion processes and filtered Poisson processes both represent the output of afilter driven by white noise, in a similar way as Gaussian processes generated by theARMA algorithm. Diffusion processes represent the output of a non-linear filterdriven by Gaussian white noise [29]. The filter is constructed so that the output hasthe target marginal PDF and an exponential covariance function. Other covariancefunctions are difficult to match [87]. Filtered Poisson processes represent theoutput of a linear filter driven by Poisson white noise [161, 164]. Poisson whitenoise can be viewed as a sequence of independent identically distributed randompulses arriving at random times given by the jump times of a Poisson process[86]. Filtered Poisson processes can match the correlation function and the firststatistical moments of the target process, but not its marginal PDF.

6.4.1 The Karhunen-Loeve decomposition

The Karhunen-Loeve decomposition of a random process is the continuouscounterpart of the decorrelation of a random vector [220]. It is derivedindependently by Karhunen, Loeve, and Kac around 1947 and introducedin the field of stochastic mechanics by Ghanem and Spanos in 1991 [78].This decomposition allows to express a random process as an infinite sum ofdeterministic functions with random coefficients. A discretized approximation ofthe random process is obtained by truncation of the infinite sum.

Derivation of the Karhunen-Loeve decomposition

The Karhunen-Loeve decomposition of the random process W(x, θ) is based onthe following eigenvalue problem:

∫

D

RW (x1,x2)f(x2) dx2 = λf(x1) (6.73)

This problem leads to the eigenvalues λk and eigenfunctions fk(x) of thecorrelation function RW (x1,x2). The eigenfunctions fk(x) are commonly referredto as Karhunen-Loeve modes. The correlation function RW (x1,x2) allows thefollowing spectral decomposition:

RW (x1,x2) =

∞∑

k=1

λkfk(x1)fk(x2) (6.74)

The Karhunen-Loeve modes fk(x) are mutually orthogonal as the correlationfunction RW (x1,x2) is symmetric. They are also normalized, so that:

∫

D

fj(x)fk(x) dx = δjk (6.75)

134 Stochastic mechanics

The Karhunen-Loeve modes fk(x) form an orthonormal basis of the vector spaceH of functions f : D → R. The random process W (x, θ) can therefore be expressedas:

W (x, θ) =

∞∑

k=1

√

λkfk(x)ξk(θ) (6.76)

This infinite sum of deterministic functions√λkfk(x) with random coefficients

ξk(θ) is the Karhunen-Loeve decomposition of the random process W (x, θ). Thecoefficients ξk(θ) follow from the projection of equation (6.76) on mode fj(x):

ξj(θ) =1√

λj

∫

D

W (x, θ)fj(x) dx (6.77)

The mean value of the random coefficients ξk(θ) is equal to:

Eξj(θ) =1√

λj

∫

D

mW (x)fj(x) dx (6.78)

Using equations (6.29), (6.74), (6.75), and (6.77), it can be shown that the randomcoefficients ξk(θ) are mutually uncorrelated:

Eξj(θ)ξk(θ) = δjk (6.79)

Discretization of random processes

A discretized approximation of the random process W (x, θ) is obtained bytruncation of the infinite sum in equation (6.76) after the M lowest order modesfx(x). The lowest order modes fx(x) correspond to the highest eigenvalues λk.This leads to:

W (x, θ) =

M∑

k=1

√

λkfk(x)ξk(θ) (6.80)

The mean square error ε2M due to the truncation is equal to:

ε2M =

∫

D

E

(

∞∑

k=M+1

√

λkfk(x)ξk(θ)

)2

dx =

∞∑

k=M+1

λ2k (6.81)

where use is made of the orthonormality of the random variables ξk(θ) and thefunctions fk(x). It can be shown that the Karhunen-Loeve modes fk(x) provide anoptimal basis for the discretization of the random process W (x, θ) with a minimalerror ε2M [78].

As in the case of a random vector, the coefficients ξk(θ) are independent Gaussianvariables if and only if the random process W (x, θ) has a Gaussian marginal PDF.

Random process modelling 135

In this case, realizations of the random variables ξk(θ) are easily obtained from a(pseudo-)random number generator. The introduction of these realizations in theKarhunen-Loeve decomposition (6.80) gives a realization of the random processW (x, θ) that can be used in a Monte Carlo simulation.

Numerical solution of the eigenvalue problem

The integral equation (6.73) is a homogeneous Fredholm equation of the secondkind. This equation can only be solved analytically in very specific cases. Ghanemand Spanos [78] proposed analytical solutions for the cases where the correlationfunction RW (x1, x2) decays exponentially or linearly with the distance |x2 − x1|.Moreover, they developed a Galerkin type procedure for the numerical solution ofthe Fredholm equation (6.73). This procedure is equivalent to the Rayleigh-Ritzmethod [19]. First, a set of P Ritz basis functions Nj(x) is chosen in the vectorspace H of functions f : D → R. The functions Nj(x) are used to calculate thefollowing P × P matrices:

[R]ij =

∫

D

∫

D

RW (x1,x2)Ni(x1)Nj(x2) dx1dx2 (6.82)

[N]ij =

∫

D

Ni(x)Nj(x) dx (6.83)

Next, the following generalized algebraic eigenvalue problem is solved:

Rv = ηNv (6.84)

This eigenvalue problems yields P eigenvalues ηk and eigenvectors vk. The eigen-values ηk are approximations of the highest P eigenvalues λk. Approximations ofthe corresponding eigenfunctions fk(x) are finally obtained as:

fk(x) =

P∑

i=1

vkiNi(x) (6.85)

Ghanem and Spanos [78] suggest to use element based polynomial shape functionsas Ritz basis functions Nj(x). In the present work, element based constant shapefunctions are used. The integrals in equations (6.82) and (6.83) are evaluated bymeans of a collocation scheme.

6.4.2 Non-Gaussian translation processes

A non-Gaussian random process W (x, θ) with a prescribed marginal PDF pW (w)and a prescribed correlation function RW (x1,x2) can be modelled as a translationprocess [83]:

W (x, θ) = g(

η(x, θ))

= F−1W

(

FG

(

η(x, θ))

)

(6.86)

136 Stochastic mechanics

where η(x, θ) is a random process with a standard Gaussian marginal PDF andg is a memoryless transformation defined in terms of the marginal CDF FW (w)of the non-Gaussian process W (x, θ) and the standard Gaussian CDF FG(η). Inorder to obtain a process with the target correlation function RW (x1,x2), thecorrelation function Rη(x1,x2) of the underlying Gaussian process η(x, θ) mustsatisfy the following equation:

RW (x1,x2) = E W (x1, θ)W (x2, θ) (6.87)

= E

g(

η(x1, θ))

g(

η(x2, θ))

(6.88)

=

∫∫

R2

g(η1)g(η2)pG

(

η1, η2|Rη(x1,x2))

dη1dη2 (6.89)

where pG

(

η1, η2|Rη(x1,x2))

denotes the joint PDF of a bivariate standardGaussian vector with correlation coefficient Rη(x1,x2). Equation (6.89) is verysimilar to equation (6.63) in subsection 6.3.2 and can be solved in the same way. Ifthe solution Rη(x1,x2) of equation (6.89) exists and is a positive definite function,it is the correlation function of the underlying Gaussian process η(x, θ).

6.5 The stochastic finite element method

This section focuses on the SFEM developed by Ghanem and Spanos [76, 78]. Theobjective of the SFEM is to characterize the response of a mechanical system withuncertain properties subjected to a deterministic or stochastic excitation. TheSFEM is developed as an extension of the deterministic finite element method.The approach followed in the SFEM is therefore also applicable to the directstiffness method, which can be considered as a special version of the finite elementmethod, using exact solutions as shape functions.

In the SFEM, the uncertain parameters are modelled as random variables orrandom processes. These variables and processes can be expressed in terms of asmall set of random variables ξk(θ) by means of the methods discussed in sections6.3 and 6.4. In this way, the random state of the system is determined by therandom variables ξk(θ). The stiffness matrix K(ξ(θ)) and the force vector F(ξ(θ))are subsequently expressed in terms of the random vector ξ(θ) that collects thevariables ξk(θ). The resulting displacement vector is then also a function of thevariables ξk(θ) and can be expressed as a response surface U(ξ(θ)). The aim ofthe SFEM is to determine this response surface as the (approximate) solution ofthe equilibrium equation:

K(ξ(θ))U(ξ(θ)) = F(ξ(θ)) (6.90)

As an example, the foundation-soil transfer functions computed in section 4.4 areconsidered. The shear modulus of the soil is now modelled as a lognormal variable

The stochastic finite element method 137

µ(θ) with a mean value mµ = 40.5 MPa and a standard deviation σµ = 8.1 MPa.The corresponding coefficient of variation equals 0.2. While this coefficient ofvariation is arbitrarily chosen, its order of magnitude corresponds to valuestypically used in the literature on stochastic soil dynamics [79, 129, 230]. Therandom shear modulus µ(θ) is expressed as a function of an underlying standardGaussian variable ξ(θ) according to equation (6.60). The transfer functions H(ω)are calculated for different values of ξ(θ) in the same way as in section 4.4. Theresults are shown in figure 6.2. The static free field response and the free fieldresponse at 100 Hz are shown. With a probability 0.95, the standard Gaussianvariable ξ(θ) takes a value in the interval [−1.96, 1.96]. In the static case, theresponse surface is very smooth in this interval. In the dynamic case, the effect ofwave propagation is clearly visible and the response surface has a complex shape.The increase of the response as the shear modulus µ increases is due to a reductionof energy dissipation, as explained in the next chapter.

(a)−3 −2 −1 0 1 2 3

−2

−1

0

1

2x 10

−10

ξ [ − ]

Dis

plac

emen

t [m

/N]

(b)−3 −2 −1 0 1 2 3

−2

−1

0

1

2x 10

−11

ξ [ − ]

Dis

plac

emen

t [m

/N]

Figure 6.2: Real (black line) and imaginary (gray line) part of the foundation-soiltransfer function H(ω) at 32 m from the center of the foundation and at (a) 0 Hzand (b) 100 Hz.

Three different methods are commonly used to determine (the statistics of) theresponse surface U(ξ(θ)): the perturbation SFEM, the spectral SFEM, and theMonte Carlo SFEM.

The perturbation SFEM is based on a Taylor series expansion of the stiffnessmatrix K(ξ(θ)), the force vector F(ξ(θ)), and the displacement vector U(ξ(θ)).

The spectral SFEM makes use of multidimensional Hermite polynomials ψi(ξ(θ))to approximate the dependence of the stiffness matrix K(ξ(θ)), the force vectorF(ξ(θ)), and the displacement vector U(ξ(θ)) on the random variables ξk(θ). Theessence of the spectral SFEM lies in the use of a Galerkin method to determinethe response surface U(ξ(θ)) in terms of the Hermite polynomials ψi(ξ(θ)).

The Monte Carlo SFEM is based on the simulation of the random variablesξk(θ) and the solution of a large number of deterministic problems of the form ofequation (6.90). This leads to a population of samples U(ξ(θ)) that represents therandom response of the system.

138 Stochastic mechanics

It is clear from figure 6.2 that the response surface U(ξ(θ)) is very difficultto approximate with either a Taylor series expansion or a Hermite polynomialexpansion if wave propagation is involved. A Monte Carlo method is thereforeused in the present work. On account of their importance in the field of stochasticmechanics, however, a brief review of the perturbation SFEM and the spectralSFEM is given in the following subsections 6.5.1 and 6.5.2. The Monte Carlomethod is discussed in subsection 6.5.3.

6.5.1 The perturbation method

The perturbation SFEM has been outlined in the early 1990s by Ghanem andSpanos [78] and Kleiber and Hien [117] and is since then applied in the field ofstructural and soil dynamics by several authors. Chakraborty and Dey [33] usethe perturbation SFEM for the analysis of a structure with an uncertain Young’smodulus subjected to earthquake loading. Yeh and Rahman [230] apply theperturbation SFEM for the site response analysis of a layered soil with an uncertainshear modulus. Van den Nieuwenhof and Coyette [220, 221] follow a modalapproach for the stochastic analysis of structures by means of the perturbationmethod, accounting for both material and geometric uncertainties.

The aim of the perturbation SFEM is to estimate the second order statistics ofthe response of a mechanical system, based on the second order statistics of theuncertain parameters in the system or the excitation. These uncertain parametersare modelled as random variables or random processes which are expressed in termsof a set of underlying random variables ξk(θ). While not absolutely necessary forthe application of the method, it is assumed here that the random variables ξk(θ)are uncorrelated (but not necessarily independent) and have a zero mean valueand a unit variance. This can easily be achieved by means of the decorrelationprocedure discussed in subsection 6.3.1 or the Karhunen-Loeve decompositiondiscussed in subsection 6.4.1. The stiffness matrix K(ξ(θ)), the force vectorF(ξ(θ)), and the displacement vector U(ξ(θ)) are subsequently expressed as aTaylor series in terms of the random variables ξk(θ):

K(ξ(θ)) = K0 +

M∑

k=1

ξk(θ)∂K

∂ξk+

1

2

M∑

j=1

M∑

k=1

ξj(θ)ξk(θ)∂2K

∂ξj∂ξk+ · · · (6.91)

F(ξ(θ)) = F0 +M∑

k=1

ξk(θ)∂F

∂ξk+

1

2

M∑

j=1

M∑

k=1

ξj(θ)ξk(θ)∂2F

∂ξj∂ξk+ · · · (6.92)

U(ξ(θ)) = U0 +

M∑

k=1

ξk(θ)∂U

∂ξk+

1

2

M∑

j=1

M∑

k=1

ξj(θ)ξk(θ)∂2U

∂ξj∂ξk+ · · · (6.93)

where M is the number of random variables ξk(θ). The coefficients in the Taylorseries expansions (6.91) and (6.92) of the stiffness matrix and the force vector can

The stochastic finite element method 139

be determined either analytically or numerically, using the finite difference method.The coefficients in the Taylor series expansion (6.93) of the response surface arefound upon the introduction of equations (6.91–6.93) in the equilibrium equation(6.90):

K0U0 +

M∑

k=1

ξk(θ)

(

K0∂U

∂ξk+∂K

∂ξkU0

)

+M∑

j=1

M∑

k=1

ξj(θ)ξk(θ)

(

1

2K0

∂2U

∂ξj∂ξk+∂K

∂ξj

∂U

∂ξk+

1

2

∂2K

∂ξj∂ξkU0

)

+ · · ·

= F0 +

M∑

k=1

ξk(θ)∂F

∂ξk+

1

2

M∑

j=1

M∑

k=1

ξj(θ)ξk(θ)∂2F

∂ξj∂ξk+ · · · (6.94)

Equating the same order polynomials on both sides of equation (6.94) yields thefollowing set of equations:

K0U0 = F0 (6.95)

K0∂U

∂ξk=∂F

∂ξk− ∂K

∂ξkU0 (6.96)

K0∂2U

∂ξj∂ξk=

∂2F

∂ξj∂ξk− ∂K

∂ξj

∂U

∂ξk− ∂K

∂ξk

∂U

∂ξj− ∂2K

∂ξj∂ξkU0 (6.97)

These equations can be solved sequentially as the right hand side is always knownfrom the previous steps. Moreover, the matrix K0 needs to be factorized onlyonce.

The solutions of equations (6.95–6.97) are introduced in equation (6.93) to obtainthe Taylor series expansion of the response surface U(ξ(θ)). In order to calculatethe mean response mU, this series is usually truncated after the second order term.Accounting for the uncorrelatedness, the zero mean value, and the unit varianceof the random variables ξk(θ), the following expression is obtained:

mU = U0 +

M∑

k=1

∂2U

∂ξ2k(6.98)

The response correlation RU is usually calculated from the first order Taylor seriesexpansion of the response surface U(ξ(θ)):

RU = U0UH0 +

M∑

k=1

∂U

∂ξk

∂UH

∂ξk(6.99)

140 Stochastic mechanics

Due to the truncation of the Taylor series expansion after the first or the secondorder term, the perturbation SFEM only yields reliable results for systems wherethe response surface U(ξ(θ)) is smooth in the vicinity of the origin. This is the caseif the relation between the uncertain parameters and the response is close to linearand if the variability in the system remains small. For the problems consideredin the present study, however, the variability of the uncertain properties is largeand the relation between the uncertain parameters and the response is stronglynon-linear, as shown in figure 6.2. The perturbation SFEM is therefore not usedin this work.

An alternative to the perturbation SFEM is the Neumann expansion SFEM,where the response surface U(ξ(θ)) is obtained from equation (6.90) by means ofa Neumann expansion of the stochastic stiffness matrix K(ξ(θ)) [78]. This methodleads to the same expression for the response surface U(ξ(θ)) as the perturbationSFEM. Therefore, the Neumann expansion SFEM is not further addressed here.

6.5.2 The spectral stochastic finite element method

The spectral SFEM has been introduced in 1991 by Ghanem and Spanos [78]. Inits original formulation, the spectral SFEM could only be used to model Gaussianuncertainties. The method has been extended afterwards to account for non-Gaussian uncertainties as well [76]. In the past decade, the spectral SFEM hasbeen used in a wide range of application fields, extending from computationalfluid dynamics [118] to molecular biology [44]. In the field of structural and soildynamics, the spectral SFEM has been used by a large number of authors. Ghioceland Ghanem [79] apply the spectral SFEM for the seismic analysis of a nuclearreactor facility. Sarkar and Ghanem use the spectral SFEM [183] for the dynamicanalysis of a structure with uncertain parameters in the mid-frequency range.Karakostas and Manolis [109] analyse the dynamic response of tunnels in a soilwith uncertain properties using Green’s functions expressed in terms of Hermitepolynomials. Liao and Li [129] calculate the site response of a soil with uncertainproperties by means of the spectral SFEM.

The spectral SFEM is based on the use of Hermite polynomials in standardGaussian variables ξ(θ) [226]. The one-dimensional Hermite polynomials Φp(ξ(θ))of order p are recursively defined as:

Φ0(ξ(θ)) = 1 (6.100)

Φ1(ξ(θ)) = ξ(θ) (6.101)

Φp+1(ξ(θ)) = ξ(θ)Φp(ξ(θ)) − pΦp−1(ξ(θ)) (6.102)

These polynomials are orthogonal:

EΦp(ξ(θ))Φq(ξ(θ)) = δpqEΦ2p(ξ(θ)) (6.103)

The stochastic finite element method 141

All Hermite polynomials Φp(ξ(θ)) of order p > 0 are orthogonal to the zero orderHermite polynomial Φ0(ξ(θ)). Consequently, the expected value of the Hermitepolynomial Φp(ξ(θ)) is given by:

EΦp(ξ(θ)) =

1 if p = 0

0 if p > 0(6.104)

The set of Hermite polynomials Φp(ξ(θ)) up to order P is called the polynomialchaos of order P . The polynomial chaos forms an orthogonal basis of the vectorspace of polynomials up to order P in a standard Gaussian variable ξ(θ). As aresult, each product of N Hermite polynomials Φpn

(ξ(θ)) from the polynomialchaos of order P can be expressed as a linear combination of the Hermitepolynomials from the polynomial chaos of order NP :

N∏

n=1

Φpn(ξ(θ)) =

NP∑

p=0

apΦp(ξ(θ)) (6.105)

The expected value of this product is given by:

E

N∏

n=1

Φpn(ξ(θ))

= a0 (6.106)

Furthermore, the Hermite polynomial chaos can be used as a basis for thepolynomial approximation of an arbitrary function f(ξ(θ)) in a standard Gaussianvariable ξ(θ):

f(ξ(θ)) ≈P∑

p=0

vpΦp(ξ(θ)) (6.107)

The coefficients vp are found by projecting the left and right hand side of equation(6.107) on the Hermite polynomial Φq(ξ):

vp =EΦp(ξ(θ))f(ξ(θ))

EΦ2p(ξ(θ))

(6.108)

The expected value in the numerator can be calculated by means of Gauss-Hermitequadrature as explained in section 6.3.2. The expected value in the denominatorfollows from equations (6.105) and (6.106).

The M -dimensional Hermite polynomials Ψi(ξ(θ)), where ξ(θ) is a randomvector containing independent standard Gaussian variables ξk(θ), are defined asproducts of M one-dimensional Hermite polynomials in the variables ξk(θ):

Ψi(ξ(θ)) =M∏

k=1

Φpik(ξk(θ)) (6.109)

142 Stochastic mechanics

For example, the two-dimensional Hermite polynomials up to order P = 2 aredefined as:

Ψ1(ξ(θ)) = 1 (6.110)

Ψ2(ξ(θ)) = ξ1(θ) (6.111)

Ψ3(ξ(θ)) = ξ2(θ) (6.112)

Ψ4(ξ(θ)) = ξ21(θ) − 1 (6.113)

Ψ5(ξ(θ)) = ξ1(θ)ξ2(θ) (6.114)

Ψ6(ξ(θ)) = ξ22(θ) − 1 (6.115)

Note that the numbering of the multidimensional Hermite polynomials starts at1, while the numbering of the one-dimensional Hermite polynomials starts at 0.Only in the one-dimensional case, the polynomials can be numbered according totheir order. In the multidimensional case, an arbitrary numbering scheme has tobe chosen. The total number Q of M -dimensional polynomials up to order P isequal to [45]:

Q =(M + P )!

M !P !(6.116)

Due to the independence of the Gaussian variables ξk(θ), the expected value of aproduct of N multidimensional Hermite polynomials Ψin(ξ(θ)) can be written as:

E

N∏

n=1

Ψin(ξ(θ))

=

M∏

k=1

E

N∏

n=1

Φpink(ξk(θ))

(6.117)

where the expected value on the right hand side can be calculated from equations(6.105) and (6.106). It follows from equation (6.117) and from the orthogonalityof the one-dimensional Hermite polynomials that the multidimensional Hermitepolynomials are also orthogonal:

EΨi(ξ(θ))Ψj(ξ(θ)) = δijEΨ2i (ξ(θ)) (6.118)

Similarly, the expected value of the multidimensional Hermite polynomial is foundas:

EΨi(ξ(θ)) =

1 if i = 1

0 if i > 1(6.119)

The set of M -dimensional Hermite polynomials Ψi(ξ(θ)) up to order P is calledthe M -dimensional polynomial chaos of order P . This polynomial chaos forms

The stochastic finite element method 143

an orthogonal basis of the vector space of M -dimensional polynomials of order Pin a standard Gaussian vector ξ(θ). It can be used as a basis for the polynomialapproximation of an arbitrary function f(ξ(θ)) in a standard Gaussian vector ξ(θ):

f(ξ(θ)) ≈Q∑

i=1

viΨi(ξ(θ)) (6.120)

The coefficients vi are found in a similar way as in the one-dimensional case:

vi =EΨi(ξ(θ))f(ξ(θ))

EΨ2i (ξ(θ))

(6.121)

In the spectral SFEM, the random variables and random processes that representthe uncertain parameters are first expressed as functions of a set of M mutuallyindependent standard Gaussian variables ξk(θ). To this end, the methods discussedin sections 6.3 and 6.4 can be used. Next, the stiffness matrix K(ξ(θ)) and the forcevector F(ξ(θ)) are approximated by means of a Hermite polynomial expansion oforder PK and the displacement vector U(ξ(θ)) is approximated by means of aHermite polynomial expansion of order PU:

K(ξ(θ)) =

QK∑

i=1

Ψi(ξ(θ))Ki (6.122)

F(ξ(θ)) =

QK∑

i=1

Ψi(ξ(θ))Fi (6.123)

U(ξ(θ)) =

QU∑

i=1

Ψi(ξ(θ))Ui (6.124)

Herein, QK and QU denote the number of M -dimensional polynomials of order PK

and PU, respectively. The coefficients Ki and Fi in equations (6.122) and (6.123)are calculated according to equation (6.121). The coefficients Ui in equation(6.124) follow from the introduction of equations (6.122–6.124) in the equilibriumequation (6.90):

QK∑

i=1

QU∑

j=1

Ψi(ξ(θ))Ψj(ξ(θ))KiUj ≈QK∑

i=1

Ψi(ξ(θ))Fi (6.125)

The equilibrium equation cannot be satisfied exactly due to the truncation of theHermite polynomial expansion of the response after QU terms. In a similar way asin the deterministic finite element method, a weighted residual approach is followedin order to minimize the truncation error. The Galerkin method is applied: the left

144 Stochastic mechanics

and right hand side of equation (6.125) are projected on the Hermite polynomialsused in the expansion of the response and their equality is enforced:

E

Ψk(ξ(θ))

QK∑

i=1

QU∑

j=1

Ψi(ξ(θ))Ψj(ξ(θ))KiUj

= E

Ψk(ξ(θ))

QK∑

i=1

Ψi(ξ(θ))Fi

(6.126)

Accounting for the orthogonality of the Hermite polynomials, equation (6.126) canbe reformulated as:

QK∑

i=1

QU∑

j=1

cijkKiUj = dkFk (6.127)

where:

cijk = E Ψi(ξ(θ))Ψj(ξ(θ))Ψk(ξ(θ)) (6.128)

dk = E

Ψ2k(ξ(θ))

(6.129)

The coefficients cijk and dk are calculated by means of equation (6.117). Equation(6.127) is a system of NQU scalar equations, where N is the number of degreesof freedom in the deterministic finite element model. If the number M of randomvariables ξk(θ) is large or the order PU of the Hermite polynomial expansion of theresponse is high, the system of equations (6.127) becomes very large and mightrequire an iterative solution [77, 162].

The coefficients Uj obtained from equation (6.127) are introduced in the Hermitepolynomial expansion (6.124) of the response in order to approximate the responsesurface U(ξ(θ)). This approximation can be used to estimate the PDF of theresponse by means of a Monte Carlo simulation of the Gaussian variables ξk(θ).The mean response mU is found in a direct way:

mU = E

QU∑

i=1

Ψi(ξ(θ))Ui

= U1 (6.130)

The response correlation RU is found analogously:

RU = E

QU∑

i=1

Ψi(ξ(θ))Ui

QU∑

j=1

Ψj(ξ(θ))UHj

=

QU∑

i=1

diUiUHi (6.131)

Compared to the perturbation SFEM, the spectral SFEM has a number ofadvantages. First, it is relatively easy to include higher order terms in the

The stochastic finite element method 145

Hermite polynomial expansions to improve the accuracy of the response surfaceapproximation. In doing so, however, the system of equations (6.127) rapidlybecomes prohibitively large. Second, the perturbation SFEM only accounts forthe second order statistics of the uncertain parameters in the model. The spectralSFEM also accounts for their PDF, albeit in an approximate way due to thetruncation of the Hermite polynomial expansions (6.122) and (6.123). Third, thespectral SFEM allows to estimate any response statistic by means of a Monte Carlosimulation of the Gaussian variables ξk(θ) in equation (6.124). The method haseven been used to calculate the failure probability of a structure, but without muchsuccess [75]. This is not surprising as in equation (6.126), the weight assigned tothe equilibrium residual for a given value of the random vector ξ(θ) is proportionalto the probability of this value. For random states of the structure that are unlikelyto occur, such as the failure state, this probability is low and, consequently, theaccuracy of the response surface approximation (6.124) is poor.

In problems governed by wave propagation, the response surface on the uncertainparameters has a complex shape and is difficult to approximate using Hermitepolynomials. In such cases, the spectral SFEM is not practical [126]. The spectralSFEM is therefore not used in the present work.

6.5.3 The Monte Carlo method

Monte Carlo simulations are widely used in many scientific disciplines, includingthe field of soil dynamics [141]. Rahman and Yeh [170] and Nour et al. [155]calculate the seismic response of a layer on rigid bedrock by means of a MonteCarlo method. Toubalem et al. [218] analyse the transfer functions of soils with astiffness that varies in the horizontal or the vertical direction by means of a MonteCarlo method, using a simplified soil model. Savin and Clouteau [184] use a MonteCarlo method to solve a problem of dynamic soil-structure interaction where thesoil exhibits random heterogeneities over a bounded domain.

In the Monte Carlo SFEM, a large number of realizations of the uncertainparameters in the system or the excitation is generated, according to theirprescribed probability distributions and correlation structure. Methods asdiscussed in sections 6.3 and 6.4 are used for this purpose. Next, a deterministicproblem is solved for each realization. This leads to a population of realizations ofthe random response that can be denoted as U(θn), where the index n refers to then-th realization. Using this population, any response statistic sU = Ef(U(θ)can be estimated as:

sU ≈ 1

N

N∑

n=1

f(U(θn)) (6.132)

where N is the number of samples in the Monte Carlo simulation. It is intuitivelyclear that the quality of this estimation improves as the number of samples Nincreases.

146 Stochastic mechanics

In order to analyse the convergence of equation (6.132) more rigorously, therandomness that underlies the generation of the samples f(U(θn)) has to betaken into account. Therefore, these samples are considered as a sequence of Nindependent identically distributed random variables Vn(θ) with a mean valuemV .The response statistic sU corresponds to the mean value mV and the estimationof sU is given by the average value VN (θ) of the samples Vn(θ):

VN (θ) =1

N

N∑

n=1

Vn(θ) (6.133)

It follows from equation (6.133) that the sample average VN (θ) itself is arandom variable. The convergence of the (random) sample average VN (θ) to the(deterministic) mean value mV is ensured by the laws of large numbers, initiallydescribed by Bernoulli in the early 18th century [195]. A distinction is madebetween the weak law of large numbers and the strong law of large numbers [65].The weak law of large numbers states that, for all ε > 0:

limN→∞

P∣

∣VN (θ) −mV

∣

∣ > ε

= 0 (6.134)

This means that, for a large enough number N of samples Vn(θ), the probabilitythat the sample average VN (θ) is not close to the mean value mV becomesarbitrarily small, but not zero. The strong law of large numbers states that:

P

limN→∞

VN (θ) = mV

= 1 (6.135)

For a large enough number N of samples Vn(θ), it is therefore almost sure (i.e. theprobability is one) that the sample average VN (θ) is close to the mean value mV .

While the strong law of large numbers ensures that equation (6.132) convergesalmost surely if the number N of samples tends to infinity, it does not allow todetermine which number N of samples is required to reach a prescribed accuracy.The central limit theorem offers a solution [42, 43, 65]. This theorem is applicableto the sequence of random variables Vn(θ) if their variance σ2

V exists. In this case,it is possible to define the following random variable:

ZN(θ) =VN (θ) −mV

σV /√N

(6.136)

The central limit theorem states that the probability distribution of the randomvariable ZN (θ) tends to the standard Gaussian distribution as N increases. As aconsequence, the probability that the sample average VN (θ) differs from the meanvalue mV by less than ε is given by:

P

∣

∣VN (θ) −mV

∣

∣ < εσV√N

= FG(ε) − FG(−ε) (6.137)

Conclusion 147

where FG(ε) is the standard Gaussian CDF. Equation (6.137) can be used toassess if the number N of samples in a Monte Carlo simulation suffices to reachthe prescribed level of accuracy. Generally, the standard deviation σV is unknown,but it can be estimated from the samples Vn(θ) as:

σ2V = EV 2(θ) −m2

V ≈ 1

N

N∑

n=1

V 2n (θ) −

(

1

N

N∑

n=1

Vn(θ)

)2

(6.138)

As an alternative to the use of equation (6.137), it is also possible to assess theconvergence of a Monte Carlo simulation by performing multiple simulation runsand comparing the results [179].

The Monte Carlo method offers a large number of advantages over the methodsconsidered in the previous subsections. The method does not involve anapproximation of the response surface and allows to estimate any response statistic,up to any level of accuracy, using deterministic solution techniques. In theliterature, the Monte Carlo method is therefore generally accepted as the preferredmethod for the validation of alternative solution techniques. However, the MonteCarlo method can be computationally expensive if a large number of realizationsis required.

In the present work, the Monte Carlo method is used to model wave propagationin a soil with uncertain properties. The convergence of the Monte Carlo simulationsis verified by comparing the results obtained from different simulation runs. Inorder to limit the computation time, the calculation is parallelized and use ismade of a high performance computing cluster. The results of the simulations arepresented in the next chapter.

6.6 Conclusion

This chapter focuses on the modelling of mechanical systems with uncertainproperties. A probabilistic approach is followed: the uncertain properties aremodelled as random variables or random processes, usually characterized bymarginal probability distributions and correlation or covariance functions.

First, the Kolmogorov probability theory is introduced to serve as a frameworkfor the description of random variables and random processes.

Next, it is shown how random vectors and random processes with prescribedmarginal probability distributions and a prescribed correlation structure can bemodelled. Random vectors are modelled by means of the Nataf transformation,which expresses a non-Gaussian vector as a function of an underlying Gaussianvector. The underlying Gaussian vector is subsequently decorrelated using theeigen decomposition of its correlation matrix. In a similar way, random processesare modelled as translation processes: non-Gaussian processes are expressed as afunction of an underlying Gaussian process. The underlying Gaussian process isdiscretized by means of its Karhunen-Loeve decomposition.

148 Stochastic mechanics

Finally, the SFEM is discussed. The SFEM is a popular method for thenumerical solution of stochastic problems in mechanics. In the SFEM, theuncertain parameters in the system or the excitation are modelled as randomvariables or processes, which are expressed in terms of a limited number ofunderlying random variables ξk(θ) by means of the methods considered in thepreceding sections. The response of the system is expressed as a response surfacein terms of the variables ξk(θ). In the perturbation SFEM and the spectral SFEM,this response surface is approximated by means of a Taylor series expansion or aHermite polynomial expansion. In the Monte Carlo SFEM, the response surfaceis randomly sampled.

For problems governed by wave propagation in a medium with uncertainproperties, it is observed that the response surface has a complex shape and isdifficult to approximate. The perturbation SFEM and the spectral SFEM are notpractical in such cases. In the next chapter, a Monte Carlo method is thereforeused to study wave propagation in a soil with random properties.

Chapter 7

Wave propagation in a soil

with uncertain properties

7.1 Introduction

This chapter focuses on wave propagation in a soil with uncertain properties. Thedynamic soil-structure interaction problem in Lincent, addressed in section 4.4, isreconsidered here. The variability of the foundation-soil transfer functions due touncertainty on the dynamic soil properties is assessed by means of a Monte Carlomethod.

Similar problems have been studied by other authors. Yeh and Rahman [230]apply the stochastic finite element method for the site response analysis of a layeredsoil modelled with finite elements. The dynamic shear modulus is modelled as aGaussian process that varies in the vertical direction. As the response is dominatedby vertically propagating shear waves, a one-dimensional model suffices. Thestochastic system equations are solved by means of a Monte Carlo simulation, aNeumann expansion, and a projection of the response on the polynomial chaos.

Ghiocel and Ghanem [79] present a procedure for the probabilistic analysis ofthe seismic soil-structure interaction problem. Finite element models are used forboth the soil and the structure. The earthquake ground acceleration, the low strainshear modulus of the soil, and the dependence of the shear modulus on the shearstrain are modelled as non-Gaussian random processes, where the shear modulusis assumed to vary only in the vertical direction. The Young’s modulus and thematerial damping of the structure are modelled as random variables characterizedby a mean value and a standard deviation. The stochastic finite element methodis used to assemble the system equations. A hybrid method is applied to solvethese equations where the projection of the response on the polynomial chaos isestimated by means of a stratified sampling technique.

Liao and Li [129] use a two-dimensional finite element model to calculate the

149

150 Wave propagation in a soil with uncertain properties

seismic response of a layer on rigid bedrock where the Young’s modulus is modelledas a random process. Absorbing boundary conditions are employed in order tofulfill Sommerfeld’s radiation conditions. The stochastic system equations aresolved by means of a projection of the response on the polynomial chaos.

Nour et al. [155] use a similar model to calculate the seismic response of a layer onrigid bedrock. The soil characteristics are modelled as bivariate random processes.The lognormal PDF is used for the dynamic shear modulus while the beta PDF isused for both the material damping ratio and the Poisson’s ratio. The stochasticresponse is calculated by means of a Monte Carlo simulation where realizations ofthe non-Gaussian soil characteristics are obtained as transformations of underlyingGaussian processes.

These authors all use the finite element method to model the soil. In thepresent work, the subdomain approach outlined in chapter 4 is followed. Thesoil is efficiently modelled with the direct stiffness method, avoiding the needfor a large finite element model with absorbing boundaries. As indicated inthe previous chapter, the perturbation stochastic finite element method and thespectral stochastic finite element method are not practical in this case due to thecomplex shape of the response function. A Monte Carlo method is therefore usedto calculate the foundation-soil transfer functions.

This chapter is subdivided into two sections. In section 7.2, the soil is modelledas a homogeneous halfspace with uncertain dynamic properties. The uncertainproperties are modelled as non-Gaussian random variables. The effect on thefoundation-soil transfer functions of an uncertain shear modulus, Poisson’s ratio,hysteretic material damping ratio, and density is considered. The aim of thissection is to emphasize the importance of the dynamic shear modulus and thematerial damping ratio in the prediction of ground vibrations.

The influence of uncertainty on the dynamic shear modulus is further studied insection 7.3, where its spatial variation is taken into account. The variation withdepth of the dynamic shear modulus is modelled as a non-Gaussian translationprocess with a prescribed marginal PDF and a prescribed covariance function.The influence of the spatial variation on the foundation-soil transfer functions isinvestigated by means of a parametric study.

A synthetic stochastic soil model is used in the present chapter: the probabilitydistribution and correlation structure of the soil properties are arbitrarilychosen. In the next chapter, the stochastic foundation-soil transfer functionsare recalculated, using an updated stochastic soil model that accounts for theinformation provided by an SASW test.

Wave propagation in a homogeneous soil with uncertain properties 151

7.2 Wave propagation in a homogeneous soil with

uncertain properties

The soil-structure interaction problem introduced in section 1.4 is reconsideredin this section. The soil is modelled as a homogeneous halfspace with uncertainproperties modelled as random variables. The foundation-soil transfer functionsare calculated by means of a Monte Carlo method. Four Monte Carlo simulationsare performed, where successively the dynamic shear modulus, the Poisson’s ratio,the material damping ratio, and the density are modelled as random variables.Based on these simulations, the relative importance of the uncertainty on thedifferent dynamic soil properties in the prediction of ground vibrations is assessed.

Obviously, the variability of the predicted transfer functions depends onthe probability distributions assumed for the uncertain soil properties. Thedistributions are chosen so that the uncertain properties can only take physicallyrealistic values. Moreover, an attempt is made to use distributions with acoefficient of variation that reflects a typical level of uncertainty for eachdynamic soil property. However, it is by definition impossible to eliminate allsubjectivity from the selection of a probability distribution, as the uncertaintyunder consideration is of the epistemic type.

In all four Monte Carlo simulations, the same mean soil model is used, with ashear modulus µ = 40.5 MPa, a Poisson’s ratio ν = 0.33, a damping ratio β = 0.03(for both shear and dilatational waves), and a density ρ = 1800 kg/m3. Thecorresponding shear and dilatational wave velocities are Cs = 150 m/s and Cp =300 m/s. In all Monte Carlo simulations, the soil-structure interaction problemis solved for 1000 realizations. The subdomain formulation used in section 4.4 isapplied, using the same parameter values, unless otherwise specified.

7.2.1 Uncertain shear modulus

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

Shear modulus [MPa]

Pro

babi

lity

dens

ity [M

Pa−

1 ]

Figure 7.1: Probability density function pµ(µ) of the random shear modulus µ(θ).

152 Wave propagation in a soil with uncertain properties

(a)0 25 50 75 100 125 150

−2

−1

0

1

2x 10

−9

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

−1

−0.5

0

0.5

1x 10

−9

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

−4

−2

0

2

4x 10

−10

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

−2

−1

0

1

2x 10

−10

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 7.2: Ten realizations of the real part of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 mfrom the center of the foundation. Darker lines correspond to higher values of therandom shear modulus µ(θ).

First, the uncertainty on the dynamic shear modulus is considered. The dynamicshear modulus is modelled as a random variable µ(θ) with the PDF pµ(µ) shownin figure 7.1. This is a lognormal PDF with a mean value mµ = 40.5 MPa anda standard deviation σµ = 8.1 MPa. The corresponding coefficient of variationequals 0.2 and the 95 % confidence region extends from 26.9 MPa to 58.5 MPa.While the coefficient of variation of the shear modulus µ(θ) is arbitrarily chosen,its order of magnitude corresponds to values typically used in the literature onstochastic soil dynamics [79, 129, 230].

Figures 7.2 and 7.3 show ten realizations of the real and imaginary part of thefoundation-soil transfer function H(ω) for four different receivers. The receiversare located at 4 m, 8 m, 16 m, and 32 m from the center of the foundation.

For low frequencies and small source-receiver distances, the response is quasi-static and force and displacement are in phase: the imaginary part of the transferfunction H(ω) vanishes and the real part reaches a local maximum. The magnitudeof this maximum is inversely proportional to the random shear modulus µ(θ) and

Wave propagation in a homogeneous soil with uncertain properties 153

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Figure 7.3: Ten realizations of the imaginary part of the foundation-soil transferfunction H(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d)32 m from the center of the foundation. Darker lines correspond to higher valuesof the random shear modulus µ(θ).

therefore differs for all realizations.At higher frequencies, waves develop in the soil, resulting in a time delay or a

phase lag between force and displacement. The phase lag depends on the ratioof the source-receiver distance and the wavelength of the dominant waves in thesoil. As the wavelength depends on the random shear modulus µ(θ), the phaselag takes a different value for all realizations. As a result, the realizations of thetransfer function H(ω) are not in phase with each other. Furthermore, the phasedifference between the realizations accumulates as the ratio of the source-receiverdistance and the wavelength increases. This phenomenon is referred to as the lossof coherence between realizations. It is clearly visible in figures 7.2 and 7.3: in thelow frequency range and for small source-receiver distances, the realizations of thetransfer function H(ω) differ in amplitude but not in phase. As the frequency orthe source-receiver distance increases, a phase difference between the realizationsoccurs. Eventually, the realizations of the transfer function H(ω) are completelyout of phase.

154 Wave propagation in a soil with uncertain properties

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Figure 7.4: Ten realizations of the modulus of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 mfrom the center of the foundation. Darker lines correspond to higher values of therandom shear modulus µ(θ).

Figure 7.4 shows ten realizations of the modulus of the transfer function H(ω). Inthe low frequency range and at small distances from the source, the modulus of thetransfer function H(ω) decreases as the random shear modulus µ(θ) increases. Thiscan easily be explained as the quasi-static impedance of the soil is proportionalto the shear modulus µ(θ). In the high frequency range and at large distancesfrom the source, the opposite is observed: the modulus of the transfer functionH(ω) increases as the random shear modulus µ(θ) increases. This is due to theattenuation of the waves in the soil, which is caused by geometrical spreadingand hysteretic material damping. The attenuation due to geometrical spreadingis independent of the wavelength, while the attenuation due to hysteretic materialdamping increases as the wavelength decreases. Waves travelling in a stiff soil(where the wavelength is large) are therefore less attenuated than waves travellingin a soft soil (where the wavelength is small).

Wave propagation in a homogeneous soil with uncertain properties 155

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Figure 7.5: Modulus of the mean value mH(ω) (thick line) and standard deviation

σH(ω) (thin line) of the foundation-soil transfer function H(ω) for differentreceivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center ofthe foundation. The soil is modelled as a homogeneous halfspace with a randomshear modulus µ(θ).

Figure 7.5 shows the modulus of the mean value mH(ω) and the standard

deviation σH(ω) of the transfer function H(ω). In the low frequency range, themodulus of the mean value mH(ω) is close to the modulus of the realizations of

the transfer function H(ω) shown in figure 7.4. The standard deviation σH(ω)equals about 0.2 times the mean value mH(ω). In the high frequency range, themean value mH(ω) of the transfer function is clearly attenuated. This is due to

the loss of coherence between different realizations of the transfer function H(ω):out-of-phase realizations average out to zero. The effect increases with the ratioof the source-receiver distance and the wavelength, due to the accumulation of thephase difference between the realizations. The attenuation of the mean transferfunction mH(ω) is not due to dissipation of energy: it is balanced by an increase ofthe transfer function’s standard deviation σH(ω). As a result, the mean transferfunction mH(ω) is no longer representative for the response observed for a single

156 Wave propagation in a soil with uncertain properties

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Figure 7.6: Mean value (thick line) and standard deviation (thin line) of themodulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.The soil is modelled as a homogeneous halfspace with a random shear modulusµ(θ).

realization and the statistics mH(ω) and σH(ω) do not allow to assess the groundvibration levels in the free field.

The assessment of the vibration levels is possible using the statisticsm|H|(ω) and

σ|H|(ω) of the modulus of the random transfer function H(ω). These statistics areshown in figure 7.6. In the entire frequency range and for all source-receiverdistances, the mean modulus m|H|(ω) of the transfer function is close to the

modulus of the realizations of the transfer function σH(ω) shown in figure 7.4.The coefficient of variation σ|H|(ω)/m|H|(ω) equals about 0.2, except for highfrequencies and large source-receiver distances, where it increases due to the effectof material damping discussed above.

Alternatively, the degree of uncertainty on the free field vibration levels can beassessed by means of a confidence region. Figure 7.7 shows the 95 % confidenceregion of the modulus of the transfer function H(ω), as well as ten realizations.

Wave propagation in a homogeneous soil with uncertain properties 157

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Figure 7.7: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.Darker lines correspond to higher values of the random shear modulus µ(θ).

This figure provides almost the same information on the response variabilityas figure 7.6, but it is presented in a way that allows for a more intuitiveinterpretation.

In order to check the convergence of the results, four additional Monte Carlosimulations are performed, all comprising 1000 realizations. For each simulation,the 95 % confidence region of the modulus of the transfer function H(ω) iscalculated. The results are shown in figure 7.8. Compared to the width ofthe confidence region, the variability of the confidence bounds remains limited,indicating that the Monte Carlo simulation has converged. The convergence of allother Monte Carlo simulations in this chapter is verified in a similar way.

158 Wave propagation in a soil with uncertain properties

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Figure 7.8: Five estimations of the 95 % confidence region of the modulus of thefoundation-soil transfer function H(ω) for different receivers, located at (a) 4 m,(b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation. The soil ismodelled as a homogeneous halfspace with a random shear modulus µ(θ).

7.2.2 Uncertain Poisson’s ratio

Second, the uncertainty on the Poisson’s ratio is addressed. The Poisson’s ratio ismodelled as a random variable ν(θ) with the PDF pν(ν) shown in figure 7.9. ThePDF pν(ν) is chosen so that the random variable 2ν(θ) follows a beta distribution.The Poisson’s ratio ν(θ) consequently takes values in the interval [0, 0.5]. Anarbitrarily chosen mean value mν = 0.33 and standard deviation σν = 0.1 areused. The corresponding coefficient of variation is 0.2 and the 95 % confidenceregion extends from 0.265 to 0.395. The probability that the Poisson’s ratio takesa value close to 0.5 is very small. The PDF pν(ν) used here is therefore onlyrealistic if the soil is dry. If it is uncertain whether the soil is dry or saturated,a bimodal PDF could be used with local maxima around ν = 0.3 (dry soil) andν = 0.5 (saturated soil). This can be implemented by means of two conditionalPDFs (for dry soil and for saturated soil) depending on a binary random variablethat determines whether the soil is dry or saturated.

Wave propagation in a homogeneous soil with uncertain properties 159

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

Poisson’s ratio [ − ]

Pro

babi

lity

dens

ity [

− ]

Figure 7.9: Probability density function pν(ν) of the random Poisson’s ratio ν(θ).

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Figure 7.10: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.Darker lines correspond to higher values of the random Poisson’s ratio ν(θ).

Figure 7.10 shows the 95 % confidence region of the modulus of the transferfunction H(ω), as well as ten realizations. Compared to the results for a soilwith an uncertain shear modulus (figure 7.7), the width of the confidence regionis small for all frequencies and source-receiver distances. The uncertainty on the

160 Wave propagation in a soil with uncertain properties

Poisson’s ratio has only a small impact on the ground vibrations in the free field.The accurate determination of the Poisson’s ratio is therefore not a priority.

7.2.3 Uncertain damping ratio

Third, the hysteretic material damping ratio is modelled as a random variable β(θ).The same value of the damping ratio is used for dilatational and shear waves. ThePDF pβ(β) of the random damping ratio β(θ) is shown in figure 7.11. This is alognormal PDF with a mean valuemβ = 0.03 and a standard deviation σβ = 0.012.This corresponds to a coefficient of variation of 0.4 and a 95 % confidence regionextending from 0.013 to 0.059. While this confidence region is rather wide, it isin accordance with the uncertainty involved in the in situ determination of thematerial damping ratio [110].

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

Damping ratio [ − ]

Pro

babi

lity

dens

ity [

− ]

Figure 7.11: Probability density function pβ(β) of the random material dampingratio β(θ).

The stochastic problem is solved by means of a Monte Carlo simulation,using 1000 realizations of the random material damping ratio β(θ). With finiteprobability, the damping ratio β(θ) takes a value close to zero, and the peakcorresponding to the Rayleigh wave in the wavenumber content of the Green’sfunctions (figure 3.4) becomes very sharp. The Green’s functions are thereforecalculated for 6000 instead of 1500 dimensionless wavenumbers between kmin

r =10−3 and kmax

r = 103. In this way, equation (3.107) ensures that the peak in thewavenumber content of the Green’s functions is properly represented for 99 % ofthe realizations.

Figure 7.12 shows the resulting 95 % confidence region and ten realizations of themodulus of the transfer function H(ω). At zero frequency, the uncertain materialdamping ratio does not affect the displacement field. The transfer functionH(ω) is therefore deterministic. As the frequency increases, the impact of thematerial damping ratio on the free field vibrations becomes stronger, especiallyat large distances from the source. As a result, the variability of the transferfunction H(ω) becomes very large. In the far field and at higher frequencies, the

Wave propagation in a homogeneous soil with uncertain properties 161

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Figure 7.12: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.Darker lines correspond to higher values of the random material damping ratioβ(θ).

confidence region even extends over multiple orders of magnitude. This confirmsthe importance of the material damping ratio in ground vibration predictions.The accurate determination of the material damping ratio of soil should thereforereceive particular attention.

7.2.4 Uncertain density

Finally, the uncertainty on the density is considered. The density is modelled asa random variable ρ(θ) with the PDF pρ(ρ) shown in figure 7.13. This PDF isarbitrarily chosen. It is a lognormal PDF with a mean value mρ = 1800 kg/m3 anda standard deviation σρ = 180 kg/m3. The corresponding coefficient of variationis 0.1 and the 95 % confidence region reaches from 1473 kg/m3 to 2178 kg/m3.

Figure 7.14 shows ten realizations and the 95 % confidence region of the modulusof the transfer function H(ω). In the static case, the uncertain density ρ(θ) has noeffect on the displacement field in the soil and the transfer function H(ω) takes adeterministic value. In the dynamic case, the density affects the wavelength of the

162 Wave propagation in a soil with uncertain properties

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1

1.5

2

2.5x 10

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Density [kg/m3]

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dens

ity [m

3 /kg]

Figure 7.13: Probability density function pρ(ρ) of the random density ρ(θ).

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Figure 7.14: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.Darker lines correspond to higher values of the random density ρ(θ).

waves in the soil. For soils with a higher density, the wavelength is smaller, the ratioof the source-receiver distance to the wavelength increases, and the attenuationdue to hysteretic material damping is stronger. This effect increases with thesource-receiver distance and the frequency. As a result, the largest variability

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 163

of the transfer function H(ω) is observed in the high frequency range and at alarge distance from the source. The variability remains limited compared to thevariability induced by the uncertainty on the dynamic shear modulus and thematerial damping ratio, however.

Subsections 7.2.1 to 7.2.4 emphasize that the most important soil properties inthe prediction of ground vibrations are the material damping ratio and the dynamicshear modulus. The Poisson’s ratio and the density are of minor importance.In the following of the thesis, however, the material damping ratio is no longerconsidered and the focus is exclusively on the dynamic shear modulus. Thedynamic shear modulus is chosen as a primary object of study in view of theavailability of well-established techniques for its determination, such as the SASWtest. However, the methodology discussed in the following is general and thereforealso applicable to the material damping ratio.

7.3 Wave propagation in an inhomogeneous soil

with an uncertain shear modulus

The dynamic soil-structure interaction problem of the previous section isconsidered again, accounting for the spatial variability of the shear modulus. Onlythe variation of the shear modulus with depth is accounted for. In this way, thesoil can be efficiently modelled by means of the direct stiffness method. Theassumption that the shear modulus does not vary in the horizontal direction ismotivated by the fact that the formation of a soil layer is governed by phenomenaaffecting large areas of land, such as erosion, sediment transport, and weatheringprocesses [67]. The shear modulus is modelled as a random process µ(z, θ) thatvaries with depth.

In subsection 7.3.1, the stochastic soil model is addressed. The properties ofthe random process µ(z, θ) are described and it is shown how this process ismodelled. In subsection 7.3.2, the stochastic foundation-soil transfer functionsH(ω) are discussed. In the following subsections, the influence of the parametersin the stochastic soil model is studied.

7.3.1 The stochastic soil model

The soil is modelled as layer with a thickness L = 16 m where the shear modulusvaries with depth on a homogeneous halfspace. In the layer 0 ≤ z ≤ L, the shearmodulus is modelled as a random process µ(z, θ). In the halfspace z ≥ L, the shearmodulus is constant and equals µ(L, θ). The influence of the depth L up to whichthe variation of the shear modulus is taken into account is studied in subsection7.3.3. The other dynamic soil properties are modelled as deterministic andspatially invariant. A Poisson’s ratio ν = 0.33, a hysteretic material damping ratioβ = 0.03 (for both shear and dilatational waves), and a density ρ = 1800 kg/m3

are used.

164 Wave propagation in a soil with uncertain properties

(a)0 20 40 60 80 100

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Pa−

1 ]

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40

60

80

Distance [m]

Cov

aria

nce

[MP

a2 ]Figure 7.15: (a) Marginal PDF pµ(µ) and (b) covariance function Cµ(z) of therandom shear modulus µ(z, θ).

The shear modulus µ(x, θ) is modelled as a stationary non-Gaussian processcharacterized by the marginal PDF pµ(µ) and the covariance function Cµ(z) shownin figure 7.15. The PDF pµ(µ) is lognormal with a mean value mµ = 40.5 MPaand a standard deviation σµ = 8.1 MPa. A Matern covariance function Cµ(z)with a smoothness parameter n = 1 and a correlation length lc = 0.25 m is used.As a result, the process µ(z, θ) has continuous realizations and is mean squarecontinuous, but not mean square differentiable.

0 1 2 3 4 5

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6

8

10

12

14

16

Cone tip resistance [MPa]

Dep

th [m

]

Figure 7.16: Cone tip resistance qc(z) recorded in Sint-Katelijne-Waver.

The choice of a covariance function Cµ(z) is difficult as it is impossible to measurethe variation of the shear modulus with depth directly. Laboratory tests onlyallow to determine the shear modulus at discrete points, while in situ tests involveaveraging of the shear modulus over a certain distance. Other soil properties, suchas the cone tip resistance qc(z) recorded in a Cone Penetration Test (CPT), canbe measured with a finer spatial resolution. While the cone tip resistance qc(z)and the dynamic shear modulus µ(z) are completely different soil properties, bothare determined by the same processes involved in the formation of a soil layer. It

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 165

is therefore assumed that both soil properties vary with depth in a similar way.The same assumption lies at the basis of empirical relationships between the conetip resistance and the dynamic shear modulus [145]. At the site in Lincent, noCPT recordings with a sufficiently fine spatial resolution are available. A profilerecorded at a different site, in Sint-Katelijne-Waver (Belgium) [159], is thereforeconsidered instead (figure 7.16). It is assumed that the variation of the conetip resistance qc(z) with depth is comparable at both sites (although there is noevidence to support this assumption). The covariance function Cµ(z) is selectedso that the realizations of the dynamic shear modulus µ(z, θ) exhibit a similarvariation with depth as the cone tip resistance qc(z) shown in figure 7.16.

(a)0 10 20 30 40 50 60

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0.2

0.3

0.4

0.5

0.6

Karhunen−Loeve mode [ − ]

Eig

enva

lue

[ − ]

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−0.6

−0.4

−0.2

0

0.2

0.4

0.6A

mpl

itude

[ −

]

Distance [m]

Figure 7.17: (a) Eigenvalues λk and (b) eigenfunctions fk(z) corresponding tothe lowest order modes in the Karhunen-Loeve decomposition of the underlyingGaussian process η(z, θ). In figure (b), darker lines represent modes of lower order.

The random shear modulus µ(z, θ) is modelled as a translation process, andhence as a memoryless transformation of an underlying Gaussian process η(z, θ).The underlying Gaussian process η(z, θ) is discretized by means of a Karhunen-Loeve decomposition. Figure 7.17a shows the eigenvalues λk corresponding tothe lowest order modes in the Karhunen-Loeve decomposition of the Gaussianprocess η(z, θ). The eigenvalues λk determine the contribution of each mode tothe random process η(z, θ). As the eigenvalues decrease for higher order modes, theKarhunen-Loeve decomposition can be truncated after a finite number M of loworder modes to obtain an approximation of the random process. A number of M =256 Karhunen-Loeve modes is used. This number is relatively high, but the impactof the order M of the Karhunen-Loeve decomposition on the computation time isnegligible. In the next chapter, where the inverse problem in the SASW methodis solved by means of a probabilistic method, this impact is large, however. Theinfluence of the orderM of the Karhunen-Loeve decomposition on the prediction ofthe foundation-soil transfer functions is therefore investigated in subsection 7.3.4.

Figure 7.17b shows that Karhunen-Loeve modes of higher order vary on a smallerspatial scale than modes of lower order. The order M of the Karhunen-Loeve

166 Wave propagation in a soil with uncertain properties

0 20 40 60 80 100

0

2

4

6

8

10

12

14

16

Shear modulus [MPa]

Dep

th [m

]

Figure 7.18: Realization of the random shear modulus µ(z, θ) with a correlationlength lc = 0.25 m.

decomposition therefore determines the smallest scale of variation represented bythe discretized random processes η(z, θ) and µ(z, θ).

Figure 7.18 shows a realization of the random shear modulus µ(z, θ). Thevariation with depth observed in this figure is similar to the variation with depthof the cone tip resistance qc shown in figure 7.16 (if the increasing trend isdisregarded).

7.3.2 The foundation-soil transfer functions

The stochastic soil-structure interaction problem is solved by means of a MonteCarlo simulation using 1000 realizations. For each realization, the interactionproblem is solved by means of the subdomain formulation introduced in chapter4, using the same parameter values as in section 4.4. The Green’s functions ofthe soil are calculated with the direct stiffness method, using 160 layer elementswith a thickness d = 0.1 m on top of a halfspace element. The convergence of theresults as a function of the layer thickness d is verified for a limited number ofrealizations.

The solution of a single soil-structure interaction problem takes about 160 s onan AMD Opteron 275 processor. In order to reduce the total computation time,the Monte Carlo simulation is performed on a high performance computing cluster,using 50 parallel calculation nodes.

Figure 7.19 shows ten realizations and the 95 % confidence region of the modulusof the foundation-soil transfer function H(ω) for different receivers. At certainfrequencies and source-receiver distances, dips are observed in the realizations ofthe transfer function H(ω). These dips are due to destructive interference ofdifferent wave patterns in the heterogeneous medium.

The width of the confidence region increases with the frequency. This is inaccordance with the general observation that the uncertainty on the transferfunctions of a mechanical system is larger in the higher frequency range. It can be

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 167

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Figure 7.19: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.The soil is modelled as an inhomogeneous halfspace with a random shear modulusµ(z, θ).

explained as follows [188]. At the soil’s surface, the motion is dominated by theRayleigh wave. The Rayleigh wave involves large displacements in a region nearthe surface with a thickness proportional to the Rayleigh wavelength λR. Theaverage soil properties in this region determine the contribution of the Rayleighwave to the response. For low frequencies, this region is large compared to thecorrelation length lc of the dynamic shear modulus µ(z, θ). For each realization,the spatial average of the shear modulus µ(z, θ) in this region is close to the meanshear modulus mµ. The spatial variation of the soil properties is therefore notresolved by the waves and the soil behaves as the mean system. As a result,the uncertainty on the transfer function H(ω) is low and the confidence regionis narrow in the low frequency range. For high frequencies, the Rayleigh wavetravels through a smaller region and the average soil properties in this regiondiffer for every realization. Consequently, the Rayleigh wave is affected by thespatial variation of the shear modulus µ(z, θ) and its contribution to the free fieldresponse is variable. This leads to a higher uncertainty on the transfer functionH(ω) and a wider confidence region in the high frequency range.

168 Wave propagation in a soil with uncertain properties

The variability of the shear modulus µ(z, θ) does not only affect the modulusof the transfer function H(ω) but also leads to a loss of coherence as discussedin section 7.2.1. As the loss of coherence is related to the phase of the transferfunction H(ω), it can not be observed in figure 7.19.

7.3.3 Influence of the domain of the random process

This subsection focuses on the influence of the depth L up to which the spatialvariation of the dynamic shear modulus µ(z, θ) is accounted for. A randomlychosen realization of the shear modulus µ(z, θ) is considered. The variation of theshear modulus below a depth L′ ≤ L is discarded and the transfer function H(ω)is calculated. Values L′ = 4 m, L′ = 8 m, and L′ = 16 m are used.

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Figure 7.20: Realization of the modulus of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m fromthe center of the foundation. The soil is modelled as an inhomogeneous halfspacewith a random shear modulus µ(z, θ) that varies up to a depth L′ = 4 m (lightgray lines), L′ = 8 m (dark gray lines), and L′ = 16 m (black lines).

The results are shown in figure 7.20. In the near field, very similar results areobtained for all values of L′. Only at very low frequencies, small differences areobserved. Low frequency waves penetrate deeply into the soil and are thereforeaffected by the spatial variation of the dynamic shear modulus µ(z, θ) at depths

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 169

below L′. In the far field, the response at the soil’s surface is influenced byreflected and refracted waves. Throughout the entire frequency range, these wavesare affected by the variation of the shear modulus at depths below L′. As aconsequence, the depth L′ has an impact on the transfer function H(ω). Thedifference between the transfer functions H(ω) obtained with the values L′ = 8 mand L′ = 16 m is limited, however. Whereas figure 7.20 only shows a singlerealization, a larger number has been examined and similar conclusions can bedrawn for each realization. A stochastic soil model with a shear modulus thatvaries up to a depth of 8 m instead of 16 m is therefore expected to yield verysimilar results to those shown in figure 7.19.

7.3.4 Influence of the order of the Karhunen-Loeve decom-

position

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This subsection concentrates on the influence of the order of the Karhunen-Loeve decomposition of the Gaussian process η(z, θ) underlying the random shearmodulus µ(z, θ). A randomly chosen realization of the shear modulus µ(z, θ) isconsidered. Only the M ′ ≤M lowest order Karhunen-Loeve modes are taken intoaccount and the transfer function H(ω) is calculated. Values M ′ = 8, M ′ = 16,and M ′ = 256 are used.

Figure 7.21 shows this realization for each value of M ′. This figure clearlydemonstrates that the order of the Karhunen-Loeve decomposition determines thesmallest scale of variation represented by the random process µ(z, θ). It also showsthat the variance of the random process µ(z, θ) is not fully accounted for if only alimited number M ′ of Karhunen-Loeve modes is used.

Figure 7.22 shows the realizations of the transfer function H(ω) obtained withKarhunen-Loeve decompositions of different order M ′. The influence of theorder M ′ is similar for all source-receiver distances. In the low frequency range,

170 Wave propagation in a soil with uncertain properties

wavelengths are large and the waves do not resolve the small scale variations ofthe soil that are represented by the higher order Karhunen-Loeve modes. Modesof order higher than 8 are not resolved and the difference between the realizationsof the transfer function H(ω) obtained with M ′ = 8, M ′ = 16, and M ′ = 256Karhunen-Loeve modes is negligible. In the high frequency range, the waves doresolve Karhunen-Loeve modes of an order higher than 8 and the realizations of thetransfer function H(ω) obtained with M ′ = 8 and M ′ = 256 are clearly different.The difference between the realizations obtained with M ′ = 16 and M ′ = 256is still very small, however. While only a single realization is shown in figures7.22 and 7.21, a larger number has been inspected and similar conclusions can bedrawn for all realizations. A stochastic soil model based on a Karhunen-Loevedecomposition of order 16 instead of 256 is therefore expected to give almost thesame results as those shown in figure 7.19.

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Figure 7.22: Realization of the modulus of the foundation-soil transfer functionH(ω) for different receivers, located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m fromthe center of the foundation. The soil is modelled as an inhomogeneous halfspacewith a random shear modulus µ(z, θ), using a Karhunen-Loeve decomposition oforder M ′ = 8 (light gray lines), M ′ = 32 (dark gray lines), and M ′ = 256 (blacklines).

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 171

7.3.5 Influence of the correlation length

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In subsection 7.3.1, the correlation length lc = 0.25 m of the random shearmodulus µ(z, θ) is determined from the variation with depth of the cone tipresistance recorded during a CPT. The influence of the correlation length lc onthe stochastic transfer function H(ω) is studied by means of two additional MonteCarlo simulations using a smaller correlation length lc = 0.125 m and a largercorrelation length lc = 0.5 m. These values are arbitrarily chosen. For both values,the covariance function Cµ(z) is shown in figure 7.23. A realization of the randomshear modulus µ(z, θ) is shown in figure 7.24. Clearly, the scale of variation of therandom process µ(z, θ) increases as the correlation length lc increases.

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Figures 7.25 and 7.26 show ten realizations and the 95 % confidence region of themodulus of the transfer function H(ω), obtained with both values of the correlationlength lc. In the case of a small correlation length lc = 0.125 m, a large fraction

172 Wave propagation in a soil with uncertain properties

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Figure 7.25: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.The soil is modelled as an inhomogeneous halfspace with a random shear modulusµ(z, θ) with a correlation length lc = 0.125 m.

of the spatial variation of the shear modulus µ(z, θ) is not resolved by the wavesin the soil. This results in a relatively narrow confidence region (figure 7.25).As the correlation length increases, the waves resolve a larger part of the spatialvariation of the shear modulus µ(z, θ) and the confidence region becomes wider(figure 7.26). Comparison of figures 7.25 and 7.26 shows that the correlation lengthlc has a considerable influence on the variability of the transfer function H(ω). Thevalue of the correlation length lc used in the stochastic soil model should thereforebe carefully selected. This is a difficult task, as the spatial variation of the shearmodulus can not be directly measured. In the present work, the variation of thecone tip resistance with depth is considered. A weakness of this approach is the lackof evidence that the dynamic shear modulus and the cone tip resistance vary withdepth in a similar way. The development of a more robust method to determinethe correlation length of the dynamic shear modulus of soil is beyond the scope ofthe present thesis, but can be the subject of further research. However, a certainlevel of subjectivity in the selection of a correlation length is unavoidable.

Wave propagation in an inhomogeneous soil with an uncertain shear modulus 173

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Figure 7.26: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.The soil is modelled as an inhomogeneous halfspace with a random shear modulusµ(z, θ) with a correlation length lc = 0.5 m.

7.3.6 Influence of the marginal PDF

Finally, the influence of the marginal PDF pµ(µ) of the random shear modulusµ(z, θ) is studied. An additional Monte Carlo simulation is performed, using arandom shear modulus µ(z, θ) with a uniform distribution pµ(µ). The mean valuemµ = 40.5 MPa and the standard deviation σµ = 8.1 MPa of the shear modulusµ(z, θ) are identical to the values used in the previous subsections. The uniformPDF pµ(µ) is shown in figure 7.27.

Figure 7.28 shows a realization of the uniformly distributed shear modulusµ(z, θ). The effect of the hard bounds imposed on the shear modulus µ(z, θ)by the uniform pµ(µ) distribution is visible.

Figure 7.29 shows ten realizations and the 95 % confidence region of the modulusof the transfer function H(ω) obtained with a uniformly distributed shear modulusµ(z, θ). The confidence region is almost indistinguishable from the confidenceregion obtained with a lognormally distributed shear modulus µ(z, θ) shown infigure 7.19. For fixed values of the mean shear modulus mµ and the standard

174 Wave propagation in a soil with uncertain properties

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deviation σµ, the influence of the marginal probability distribution pµ(µ) on the

variability of the transfer function H(ω) is very small.

7.4 Conclusion

In this chapter, wave propagation in a soil with uncertain properties is modelledby means of a Monte Carlo method. The soil-structure interaction problemintroduced in section 1.4 is reconsidered and the effect of uncertainty on thedynamic shear modulus, the Poisson’s ratio, the material damping ratio, and thedensity is investigated. It is observed that the uncertainty on the dynamic shearmodulus and the material damping ratio has a large impact on the variability ofthe foundation-soil transfer functions. Well-established techniques to determinethe dynamic shear modulus are already available, such as the SASW test and theSCPT. Further research should therefore focus on the development or refinementof new or existing methods to determine the material damping ratio.

Conclusion 175

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Figure 7.29: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the foundation-soil transfer function H(ω) for different receivers,located at (a) 4 m, (b) 8 m, (c) 16 m, and (d) 32 m from the center of the foundation.The soil is modelled as an inhomogeneous halfspace with a random shear modulusµ(z, θ) with a uniform probability distribution pµ(µ).

In addition, the uncertain spatial variation of the shear modulus is taken intoaccount. The shear modulus is modelled as a random process with a certaincorrelation length lc. In the low frequency range, the wavelength of the wavesin the soil is large compared to the correlation length lc. As a result, the wavesdo not resolve the spatial variation of the shear modulus, and the variability ofthe foundation-soil transfer functions is limited. In the high frequency range, thewaves do resolve the spatial variation of the shear modulus and the variability ofthe foundation-soil transfer functions increases. It is observed that the correlationlength lc has a large influence on the variability of the foundation-soil transferfunctions, while the influence of the marginal PDF of the shear modulus is limited.

As the simulations in this chapter are based on a synthetic stochastic soil profile,no comparison is made with the experimental data presented in section 1.4. Inthe next chapter, the SASW method is used to update the stochastic soil profile.The uncertain foundation-soil transfer functions are subsequently recalculated andcompared with the experimental data.

176 Wave propagation in a soil with uncertain properties

Chapter 8

Stochastic soil

characterization by means of

the SASW test

8.1 Introduction

In this chapter, the inverse problem in the SASW method is reconsidered in aprobabilistic framework. In the inverse problem, the dynamic soil properties areidentified from the experimental dispersion curve. The direct stiffness method[113] is used to calculate the theoretical dispersion curve of a soil with a givenstiffness profile. The stiffness profile is iteratively modified in order to minimize amisfit function defined as the distance between the theoretical and experimentaldispersion curves. However, the dispersion curve is insensitive to variations of thesoil properties on a small spatial scale or at a large depth [189]. The information onthe soil properties provided by the dispersion curve is therefore limited. As a result,the solution of the inverse problem is non-unique: the soil profile obtained with aclassical deterministic optimization scheme is only one of the profiles that fit theexperimental data. Alternative solution methods have therefore been developedto identify the ensemble of soil profiles fitting the experimental data [182].

Several authors assume that the relation between the shear modulus and thedispersion curve is only weakly non-linear and that the uncertainty on theexperimental dispersion curve is small [124, 210]. In this way, the inverse problemcan be linearized in the vicinity of a reference solution, which is obtained usinga local or global optimization scheme. The covariance function of the solutionthen follows directly from the covariance function of the measurement data. Inthis approach, the ensemble of acceptable soil profiles is characterized by a meanprofile and a covariance function.

177

178 Stochastic soil characterization by means of the SASW test

If the uncertainty on the measurement data increases or if the problem is morethan weakly non-linear, the misfit function may be complex and even multimodal.In such cases, the reference solution loses its meaning as single most probablesoil profile and the linearization of the inverse problem may lead to a substantialunderestimation of the variance of the solution [182]. The only robust techniqueto assess the uncertainty on the soil profile then is a Monte Carlo inversion methodwhere an ensemble of acceptable soil profiles is generated. An overview of MonteCarlo inversion methods in the field of geophysics is given by Sambridge andMosegaard [182].

Conceptually, Monte Carlo inversion methods are relatively simple. A largenumber of soil profiles is randomly generated. For each profile, the correspondencewith the experimental data is assessed. Only the profiles fitting the experimentaldata are accepted. However, the direct application of a Monte Carlo inversionmethod is not practical: the probability of generating an acceptable soil profileis very small. Therefore, soil profiles are generated in a controlled way. Theinversion procedure is started with an acceptable soil profile obtained from aclassical deterministic inversion scheme. Each subsequent profile is generatedthrough a small random perturbation of the previous profile. The acceptancerate is determined by the magnitude of the perturbations: smaller perturbationslead to a higher acceptance probability. This technique is called a Markov chainMonte Carlo method as the sequence of accepted samples constitutes a Markovchain. A Markov chain is a sequence of random vectors ξi(θ) where each vectorξi(θ) only depends on the previous vector ξi−1(θ) [179, 214].

Markov chain Monte Carlo methods are used in a wide variety of scientificfields. Applications range from the calculation of failure probabilities in reliabilityanalysis [15], over the characterization of social networks based on a small numberof observations [146], to the reconstruction of ancestral relationships amongdifferent species of organisms in the field of evolutionary biology [125]. All theseapplications are characterized by a very small probability of generating a sampleof interest, such as a sample for which failure occurs in a reliability analysis. Theuse of a classical Monte Carlo method is consequently unfeasible and Markov chainMonte Carlo methods are therefore resorted to.

Monte Carlo inversion techniques are usually applied in the framework of aBayesian updating scheme [21]. The Bayesian approach combines the informationavailable before the experiment (the prior information) with the informationprovided by the experimental data. The prior information is first used to assign aprior probability ρξ(ξ) to every conceivable soil profile characterized by a vectorξ. The prior PDF ρξ(ξ) reflects the degree of belief that a soil profile is the trueprofile, accounting for the prior information but not for the experimental data. Alikelihood function Lξ(ξ) is subsequently defined that reflects the degree to whicha given profile ξ fits the experimental data. This function is a PDF specifyingthe probability that the experimental data are observed for each conceivable soilprofile. The posterior probability σξ(ξ) of each soil profile is finally calculated asthe (normalized) product of the prior probability ρξ(ξ) and the likelihood Lξ(ξ).

Introduction 179

The posterior PDF σξ(ξ) reflects the degree of belief that the profile characterizedby the vector ξ is the true profile, accounting for both the prior information andthe experimental data. Usually, it is impossible to obtain an explicit expression forthe posterior PDF σξ(ξ). In such cases, a Markov chain Monte Carlo method canbe used to obtain an ensemble of samples distributed according to the posteriorPDF σξ(ξ).

Mosegaard and Tarantola [150] follow a Bayesian approach to solve a geophysicalproblem where the density of the soil is derived from the gravity at the soil’ssurface. The soil is modelled as a layered medium. A prior PDF is formulated forthe thickness and for the density of each layer. This prior stochastic soil model issampled by means of a Markov chain Monte Carlo method. For each sample, thegravity at the soil’s surface is calculated. Next, the Metropolis rule [98, 148] isused to accept or to reject the sample, depending on the correspondence with theexperimental data. This leads to an ensemble of soil profiles distributed accordingto the posterior probability distribution.

Shapiro and Ritzwoller [192] apply a similar technique for the identification of theshear wave velocity of the earth’s crust and upper mantle. The experimental dataconsist of a large set of Rayleigh and Love wave dispersion curves collected duringearthquakes. For each sample drawn from the prior model, the experimental dataare compared with the theoretical dispersion curves, which are efficiently calculatedby means of a Taylor expansion around a reference solution. Finally, the ensembleof acceptable models is inspected for features that occur in every member of theensemble. These features follow with high probability from the experimental dataand are referred to as persistent.

Beaty et al. [22] use a simulated annealing procedure to derive the shear modulusof shallow soil layers from the fundamental and higher Rayleigh mode dispersioncurves. The simulated annealing method is a global optimization method. Theprocedure is similar to the Markov chain Monte Carlo method used by Mosegaardand Tarantola [150], but the likelihood function and the random step size aregradually modified as the algorithm proceeds, in order to arrive at the globalminimum of the misfit function. Beaty et al. [22] also apply the method with afixed likelihood function and step size to estimate the variance of the identifiedsoil profile.

In this chapter, a Bayesian approach is followed to invert the experimentaldispersion curve of the soil at the site in Lincent determined in section 5.3. Thefocus is on the identification of the dynamic shear modulus of the soil, but thesame methodology can be used to identify the material damping ratio.

First, a prior stochastic model is constructed for the dynamic shear modulus.The variation of the shear modulus with depth is modelled as a randomprocess characterized by a marginal PDF and a covariance function, as in theprevious chapter. Second, a likelihood function is formulated that reflects thecorrespondence between the experimental dispersion curve and the theoreticaldispersion curve for every soil profile. Third, a Bayesian approach is followed toderive the posterior stochastic soil model from the prior model and the likelihood

180 Stochastic soil characterization by means of the SASW test

function. The posterior model is sampled by means of a Markov chain MonteCarlo method. This leads to an ensemble of acceptable soil profiles that reflectsthe uncertainty on the SASW results. Power wavelet spectra of the prior and theposterior shear modulus are compared to assess the resolution of the SASW test interms of depth and spatial scale. Finally, the ensemble of acceptable soil profilesis used in a Monte Carlo simulation where the foundation-soil transfer functionsare computed. The results are compared with the experimental transfer functionspresented in section 1.4. The variability of the transfer functions is consideredin order to assess the robustness of the SASW test for the prediction of groundvibrations.

8.2 The prior model

The prior stochastic soil model represents the information on the soil propertiesthat is available before the SASW test is performed. A prior model similar to thestochastic soil model in section 7.3.1 is used. The shear modulus is assumed tovary only in the vertical direction and is modelled as a random process µ(z, θ).The other dynamic soil properties (Poisson’s ratio, damping ratio, and density)are assumed to be known a priori as the focus in the present study is on theidentification of the dynamic shear modulus. A Poisson’s ratio ν = 0.33, ahysteretic material damping ratio β = 0.03 (for both shear and dilatational waves),and a density ρ = 1800 kg/m3 are used.

The shear modulus µ(z, θ) is modelled as a stationary non-Gaussian processcharacterized by the marginal PDF pµ(µ) and the covariance function Cµ(z) shownin figure 8.1 The PDF pµ(µ) is a uniform PDF ranging from 10.1 MPa to 162 MPa.This range corresponds to a minimal shear wave velocity of 75 m/s and a maximalshear wave velocity of 300 m/s, which is typical for shallow soil layers in Belgium.The covariance function Cµ(z) is a Matern covariance function with a smoothnessparameter n = 1. As a consequence, the process µ(z, θ) has continuous realizationsand is mean square continuous, but not mean square differentiable. The correlationlength lc = 0.25 m is chosen in the same way as in the previous chapter. Thisvalue is relatively small (compared to the wavelength of the waves in the soil). Asa result, the prior model contains distinct variations of the shear modulus on asmall spatial scale. The comparison with the resulting small scale variations inthe posterior model allows to assess the resolution of the SASW test.

The domain of the random process µ(z, θ) is restricted to the finite interval 0 ≤z ≤ L. The depth L is one of the parameters determining the number of Karhunen-Loeve modes needed to model the random process µ(z, θ), and hence the numberof random variables ξk(θ) in the stochastic model. This number determines theconvergence rate of the Markov chain Monte Carlo method described in section 8.4.The depth L is therefore kept to a minimum. It is determined from a preliminarysimulation of the foundation-soil transfer functions H(ω) using the prior stochasticsoil model. In a similar way as in subsection 7.3.3, the depth is determined up to

The prior model 181

(a)0 50 100 150 200

0

2

4

6

8x 10

−3

Shear modulus [MPa]

Pro

babi

lity

dens

ity [M

Pa−

1 ]

(b)0 0.5 1 1.5 2

0

500

1000

1500

2000

Distance [m]

Cov

aria

nce

[MP

a2 ]Figure 8.1: (a) Prior marginal PDF pµ(µ) and (b) prior covariance function Cµ(z)of the soil’s dynamic shear modulus µ(z, θ).

which the spatial variation of the shear modulus µ(z, θ) affects the foundation-soiltransfer functions H(ω) in the frequency range of interest. The frequency rangeof interest extends from 20 Hz to 150 Hz in the near field and from 20 Hz to 100 Hzin the far field. In this range, the coherence of the measured force and responseis high and the measured transfer functions H(ω) can be used as a reference toverify the numerical predictions. For frequencies above 20 Hz, it is observed fromthe preliminary simulation that the impact of the variation of the shear modulusbelow 6 m is limited. As a consequence, it is unnecessary to identify the variation ofthe dynamic shear modulus below this depth or to include it in the prior stochasticsoil model. A value L = 6 m is therefore used.

The shear modulus µ(z, θ) is modelled as a translation process:

µ(z, θ) = F−1µ

(

FG

(

η(x, θ))

)

(8.1)

where Fµ(µ) and FG are the marginal CDF of the random shear modulus µ(z, θ)and the standard Gaussian CDF, respectively. The underlying Gaussian processη(z, θ) is discretized by means of a Karhunen-Loeve decomposition:

η(z, θ) =M∑

k=1

√

λkfk(z)ξk(θ) (8.2)

where fk(z) and λk are the eigenfunctions and eigenvalues of the correlationfunction Cη(z) of the process η(z, θ). In this way, the random state of the priorstochastic soil model is uniquely determined by a set of independent standardGaussian variables ξk(θ).

The order M of the Karhunen-Loeve decomposition is chosen in a similar way asthe depth L, following the methodology used in subsection 7.3.4. For frequenciesbelow 150 Hz, it is observed that the Karhunen-Loeve modes of order higher than16 are (almost) not resolved by the waves in the soil. Only the first M = 16Karhunen-Loeve modes are therefore withheld in the prior model.

182 Stochastic soil characterization by means of the SASW test

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [MPa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

0

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 8.2: (a) Ten realizations of the shear modulus µ(z, θ) drawn from the priorstochastic soil model and (b) corresponding theoretical dispersion curves CT

R (ω)(gray lines), compared with the experimental dispersion curve CE

R(ω) (black dots).

The random variables ξk(θ) that determine the state of the prior stochastic soilmodel follow a multivariate standard Gaussian probability distribution:

ρξ(ξ) =1

(2π)M/2exp

(

−‖ξ‖2

2

)

(8.3)

where ‖ξ‖ is the L2-norm of the vector ξ. The PDF ρξ(ξ) characterizes the priorstochastic soil model and is therefore referred to as the prior PDF.

Figure 8.2 shows ten realizations of the shear modulus µ(z, θ) drawn from theprior stochastic soil model, as well as the corresponding theoretical dispersioncurves CT

R (ω). The theoretical dispersion curves CTR (ω) are derived from the

wavenumber content of the Green’s function uGzz(z

′ = 0, kr, z = 0, ω) as describedin section 5.4. The jumps in the dispersion curves indicate the dominance of higherorder surface modes. If the theoretical dispersion curves CT

R (ω) are compared withthe experimental curve CE

R(ω) obtained in section 5.3, it is obvious that they donot correspond. This is not surprising as the experimental data are not yet takeninto account.

8.3 The likelihood function

For every soil profile in the prior model, the misfit between the theoretical disper-sion curve CT

R (ω) and the experimental dispersion curve CER(ω) is characterized by

the likelihood function Lξ(ξ). The likelihood function Lξ(ξ) is a PDF specifyingthe probability that the experimental data CE

R(ω) are observed for a given state ξof the system. This probability depends on the uncertainty on the experimental

The likelihood function 183

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [Mpa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

100

120

140

160

180

200

220

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 8.3: (a) Two realizations of the random shear modulus µ(z, θ) and(b) corresponding theoretical dispersion curves CT

R (ω), compared with theexperimental dispersion curve CE

R(ω) (black dots) and the bounds imposed by thelikelihood function Lξ(ξ) (gray area). The black line corresponds to an acceptablesoil profile, the gray line to an unacceptable soil profile.

dispersion curve. In the present study, the likelihood function Lξ(ξ) is defined as:

Lξ(ξ) =

0 if maxω

∣

∣CTR (ω) − CE

R(ω)∣

∣ > ∆CR

12∆CR

if maxω

∣

∣CTR (ω) − CE

R(ω)∣

∣ ≤ ∆CR

(8.4)

where ∆CR = 5 m/s. Equation (8.4) implies that the error on the experimentaldispersion curve is certainly smaller than ∆CR = 5 m/s. All soil profiles for whichthe deviation of the theoretical dispersion curve CT

R (ω) from the experimentaldispersion curve CE

R(ω) does not exceed the threshold value ∆CR are acceptableand equally likely, while the other profiles are unacceptable.

As an example, figure 8.3 shows two realizations of the random shear modulusµ(z, θ), as well as the corresponding theoretical dispersion curves CT

R (ω). Thesecurves are compared with the experimental dispersion curve CT

R (ω) and thebounds imposed by the likelihood function Lξ(ξ). The theoretical dispersion curveCT

R (ω) corresponding to the first realization of the shear modulus µ(z, θ) is withinbounds over the entire frequency range where the experimental dispersion curveis determined. This realization is therefore acceptable. The theoretical dispersioncurve CT

R (ω) corresponding to the second realization of the shear modulus µ(z, θ)is out of bounds around 30 Hz and above 110 Hz, where the deviation from theexperimental dispersion curve CE

R(ω) exceeds the threshhold value ∆CR = 5 m/s.As a consequence, this realization is not acceptable.

The choice of this likelihood function is rather arbitrary and subjective.Alternatively, a likelihood function can be formulated on the basis of a morerigorous estimation of the variability of the experimental dispersion curve. O’Neill[158] presents a numerical study of the influence of various sources of uncertaintyon the experimental dispersion curve, such as the position and tilt of the sensors.

184 Stochastic soil characterization by means of the SASW test

Marosi and Hiltunen [142] determine the variance of the experimental dispersioncurve using a large sample of experimental data collected from two test sites andfind a coefficient of variation of 2 %. Lai et al. [124] follow a similar approach andfind coefficients of variation ranging from 1.1 % to 5 % in the low frequency rangeand from 0.23 % to 1.25 % in the high frequency range.

8.4 The Bayesian updating scheme

In this section, a Bayesian approach is followed to transform the prior soil modelinto a posterior soil model, using the likelihood function Lξ(ξ). The posterior soilmodel is characterized by a posterior PDF σξ(ξ), which is formally defined as:

σξ(ξ) = kρξ(ξ)Lξ(ξ) (8.5)

The normalization constant k is introduced to ensure that the posterior PDFσξ(ξ) integrates to one. Equation (8.5) can not be used to obtain a closed-formexpression for the posterior PDF σξ(ξ) as the likelihood Lξ(ξ) is not explicitlyknown. Therefore, the posterior PDF σξ(ξ) is not calculated, but a Markov chainMonte Carlo inversion method is applied instead to obtain a population of soilprofiles distributed according to the posterior PDF σξ(ξ).

In the literature, a wide variety of algorithms is available to construct a Markovchain that follows a prescribed but not explicitly known probability distribution[179]. The most popular is the Metropolis-Hastings algorithm [98, 148], whichis used in the present work. The Metropolis-Hastings algorithm proceeds asfollows. First, a candidate ξ′i+1 for the next state ξi+1 is randomly generatedas a perturbation of the current state ξi using a conditional PDF q(ξ′i+1|ξi). ThisPDF determines the probability that the chain moves from the state ξi to the stateξ′i+1 and is referred to as the proposal distribution. Next, the Metropolis-Hastingsacceptance probability r(ξi, ξ

′i+1) is calculated as:

r(ξi, ξ′i+1) = min

σξ(ξ′i+1)

σξ(ξi)

q(ξi|ξ′i+1)

q(ξ′i+1|ξi), 1

(8.6)

where the factor q(ξi|ξ′i+1)/q(ξ′i+1|ξi) vanishes if the proposal distribution q is

symmetric, i.e. if the transition from ξi to ξ′i+1 and the transition from ξ′i+1 toξi are equally probable. This is the case in the original Metropolis algorithm.Finally, the candidate ξ′i+1 for the next state ξi+1 is accepted with probabilityr(ξi, ξ

′i+1). To this end, a realization u of an auxiliary random variable U(θ) with

a uniform probability distribution between 0 and 1 is generated. The candidateξ′i+1 is accepted if and only if u < r(ξi, ξ

′i+1). If the candidate is rejected, the

next state ξi+1 is set equal to the current state ξi. This procedure results in asequence of soil profiles characterized by the vectors ξi in the Markov chain, whichare distributed according to the posterior PDF σξ(ξ).

The Bayesian updating scheme 185

Following the approach proposed by Mosegaard and Tarantola [150], candidatesare accepted in two stages. In the first stage, the acceptance probability r1(ξi, ξ

′i+1)

is calculated from the prior PDF ρξ(ξ) as:

r1(ξi, ξ′i+1) = min

ρξ(ξ′i+1)

ρξ(ξi)

q(ξi|ξ′i+1)

q(ξ′i+1|ξi), 1

(8.7)

The likelihood function Lξ(ξ) is accounted for in the second stage, using theacceptance probability r2(ξi, ξ

′i+1):

r2(ξi, ξ′i+1) = min

Lξ(ξ′i+1)

Lξ(ξi), 1

(8.8)

It can be proven that this two-stage approach is equivalent to the use of theacceptance probability r(ξi, ξ

′i+1) defined in equation (8.6) [150]. The numerical

cost of the two-stage approach is smaller as the likelihood Lξ(ξ′i+1) does not have

to be computed for candidate states that have already been rejected in the firststage. In this way, the number of theoretical dispersion curves to calculate isreduced.

In the present analysis, the Markov chain is started at a state ξ1 for which thelikelihood Lξ(ξ1) = 1 and stopped after N = 106 steps. The proposal densityq(ξ′i+1|ξi) is defined as a Gaussian PDF centered around the current state ξi:

q(ξ′i+1|ξi) =1

(2π)M/2σMqexp

(

−∥

∥ξ′i+1 − ξi∥

∥

2

2σ2Mq

)

(8.9)

The standard deviation σq determines the step size in the Markov chain. A largerstep size results in a faster exploration of the support of the posterior PDF σξ(ξ),but also in a lower acceptance rate of candidate states. A standard deviationσq = 0.08 is used, resulting in an acceptance rate of about 0.2.

Figure 8.4 shows the shear modulus and the theoretical dispersion curve for tensuccessive states in the Markov chain. Due to the relatively small step size σq =0.08 used to generate new candidate states from the previous state, the differencebetween the realizations of the shear modulus shown in figure 8.4a remains limited.As a result, the difference between the corresponding theoretical dispersion curvesin figure 8.4b is also limited. As the algorithm proceeds, the theoretical dispersioncurve stays close to the experimental curve, so that a relatively large fraction ofthe candidate states is accepted.

The proposal density q(ξ′i+1|ξi) defined in equation (8.9) assigns a positiveprobability to any subset of the vector space of random variables ξk(θ). TheMarkov chain can therefore proceed in a single step from the current state ξi toany other state. As a result, the chain can be proven to converge to the posteriorPDF σξ(ξ) for every initial state ξ1 [179, 214]. The convergence of the chain afteran infinite number of steps is therefore guaranteed, but it is very difficult to assess

186 Stochastic soil characterization by means of the SASW test

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [Mpa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

100

120

140

160

180

200

220

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 8.4: (a) Shear modulus and (b) dispersion curve for ten successive candidatestates in the Markov chain. Black lines correspond to acceptable candidates, graylines correspond to unacceptable candidates.

the convergence of a chain truncated after a finite number of steps. Robert andCasella [179] suggest to monitor the convergence of averages and the convergenceto independent sampling. Both are explained and verified in the following.

The average mξk(n) and standard deviation σξk

(n) of the first n samples of therandom variables ξk in the Markov chain are calculated as follows:

mξk(n) =

1

n

n∑

i=1

ξik (8.10)

σξk(n) =

√

√

√

√

1

n

n∑

i=1

ξ2ik −m2ξk

(n) (8.11)

The results are shown in figure 8.5. Both statistics mξk(n) and σξk

(n) remainapproximately constant after 2×105 samples, indicating that the truncated Markovchain has converged.

The convergence to independent sampling is assessed through the correlationcoefficients cξk

(∆i) of successive states in the Markov chain:

cξk(∆i) =

1

σ2ξk

(N)

(

1

N − ∆i

N−∆i∑

i=1

ξikξ(i+∆i)k −m2ξk

(N)

)

(8.12)

The results are shown in figure 8.6. The correlation between samples vanishes afterabout 4000 steps, suggesting that two samples in the Markov chain are mutuallyindependent if they are 4000 or more steps apart from one another. The chaintherefore contains about 106/4000 = 250 mutually independent samples.

The number of samples required for convergence is high and, consequently, thecomputational cost to construct the chain is large. The calculation is performedon a single AMD Opteron 150 processor in about 100 h. In order to reduce the

The Bayesian updating scheme 187

(a)0 2 4 6 8 10

x 105

−2

−1

0

1

2

Number of samples [ − ]

Mea

n va

lue

[ − ]

(b)0 2 4 6 8 10

x 105

0

0.5

1

1.5

Number of samples [ − ]

Sta

ndar

d de

viat

ion

[ − ]

Figure 8.5: (a) Mean and (b) standard deviation of the random variables ξk(θ) inthe Markov chain. Darker lines correspond to lower order Karhunen-Loeve modes.

0 2000 4000 6000 8000 10000−0.2

0

0.2

0.4

0.6

0.8

1

Number of samples [ − ]

Cor

rela

tion

[ − ]

Figure 8.6: Correlation coefficient cξk(∆i) of the random variables ξk(θ) in the

Markov chain. Darker lines correspond to lower order Karhunen-Loeve modes.

computation time, it is possible to construct multiple chains in parallel on a highperformance computing cluster. If the parallel chains start from the same point ξ1,the first samples in all chains are concentrated in the same neighbourhood, whichleads to biased results. The first part of the chains should therefore be discarded.This part is commonly referred to as the burn-in period. The number of burn-initerations required for the chains to become mutually independent can be derivedfrom the correlation coefficients cξk

(∆i) shown in figure 8.6.Another option to reduce the computation time is the use of a directional

Metropolis-Hastings algorithm. Such an algorithm favors large steps in directionsfor which the expected acceptance probability is high. The aim is to acceleratethe exploration of the support of the posterior PDF σξ(ξ) with a minimal effecton the acceptance rate. In the present analysis, a different proposal densityq(ξ′i+1|ξi) could be used to provoke larger steps in the direction of the variablesξk(θ) corresponding to the Karhunen-Loeve modes of higher order. These modeshave a minimal impact on the dispersion curve [189]. Starting from an acceptable

188 Stochastic soil characterization by means of the SASW test

(a)0 25 50 75 100 125 150 175 200

0

1

2

3

4

5

6

Shear modulus [MPa]

Dep

th [m

]

(b)0 25 50 75 100 125 150 175 200

0

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 8.7: (a) Ten realizations of the shear modulus µ(z, θ) drawn from theposterior stochastic soil model and (b) corresponding theoretical dispersion curvesCT

R (ω) (gray lines), compared with the experimental dispersion curve CER(ω) (black

dots).

soil profile, a (large) perturbation of the contribution of these modes to the randomshear modulus µ(z, θ) is therefore unlikely to yield an unacceptable soil profile. Asan alternative, it is possible to adaptively update the proposal density during thecourse of the simulation in an effort to maximize both the exploration speed andthe acceptance rate [61].

8.5 The posterior model

The Markov chain constructed in the previous section represents the posteriorstochastic soil model. The study of the posterior model and the comparison withthe prior model allows the estimation of the resolution of the SASW test. In thissection, the resolution of the SASW test is assessed in terms of depth and spatialscale.

Figure 8.7a shows ten realizations of the shear modulus µ(z, θ) selected fromthe Markov chain. These realizations are separated by more than 4000 stepsand are therefore considered as mutually independent. The correspondingtheoretical dispersion curves CT

R (ω) are shown in figure 8.7b and compared withthe experimental dispersion curve CE

R(ω). The difference between the theoreticaland experimental curves does not exceed the threshold value ∆CR imposed viathe likelihood function Lξ(ξ). As a result, all soil profiles in figure 8.7a fit theexperimental data relatively well. The variability of the profiles is large, however.Below a depth of 2 m, or about 0.2 times the largest wavelength in the experimentaldispersion curve, only a small reduction of the variability is observed compared tothe prior model (figure 8.2a). Hence, the resolution of the SASW test below thisdepth is poor. A similar observation has been made by Beaty et al. [22].

The realizations of the posterior shear modulus µ(z, θ) shown in figure 8.7aexhibit pronounced variations on a small spatial scale compared to the measured

The posterior model 189

−4 −2 0 2 4−1.5

−1

−0.5

0

0.5

1

1.5

z [ − ]

ψ [

− ]

Figure 8.8: Real (black line) and imaginary (gray line) part of a fourth order Paulwavelet ψ(z).

Rayleigh wavelengths. These variations depend to a large extent on the smallscale variations incorporated in the prior soil model. It is therefore informativeto compare the prior and the posterior variability of the shear modulus µ(z, θ) ondifferent spatial scales. This is possible via the random variables ξk(θ). In the priormodel, the random variables ξk(θ) are mutually independent Gaussian variableswith unit standard deviation. In the posterior model, the variables ξk(θ) are nolonger independent or Gaussian, and their standard deviation is given by the finalvalue of the curves in figure 8.5b. Hence, the Bayesian updating scheme leadsto a reduction of the variance, especially for the variables ξk(θ) corresponding tothe lowest order Karhunen-Loeve modes. These modes represent the large scalevariations of the shear modulus µ(z, θ). In an SASW test, large scale variations ofthe shear modulus µ(z, θ) are therefore better resolved than small scale variations.The limited resolution of the SASW test is explained as follows: the phase velocityof the Rayleigh wave depends on the spatial average of the soil properties in aregion near the surface with a thickness proportional to the Rayleigh wavelength.Small scale variations of the soil properties are averaged out and have no effect onthe dispersion curve.

The small scale variations in the posterior soil model have also been observedby other authors addressing similar problems with Monte Carlo inversion methods[182]. Some authors consider the occurrence of these small scale variations as adrawback of Monte Carlo inversion methods as they are believed to be physicallyunrealistic. This argument does not hold in the present case: soil profiles exhibitingvariations that are (a priori) considered as physically unrealistic should be assigneda zero (or very small) prior probability through the selection of a proper priorcorrelation function Cµ(z), with a sufficiently large correlation length lc.

It is impossible to separate the influence of the depth and the spatial scale onthe resolution of the SASW test. Wavelet analysis provides a tool to reveal thecombined influence of both. The wavelet power spectrum |W (z, kz)|2 of the priorand the posterior shear modulus µ(z, θ) is calculated following the guidelines of

190 Stochastic soil characterization by means of the SASW test

(a) (b)

Figure 8.9: (a) Prior and (b) posterior wavelet power spectrum |W (z, kz)|2 of thesoil’s dynamic shear modulus µ(z, θ). The hatched area is the cone of influence.

Torrence and Compo [217]. For each depth z and each wavenumber kz , the randomshear modulus µ(z, θ) is projected on a fourth order Paul wavelet ψ(z) (figure8.8) that is translated and dilated so that it is centered around z in the spacedomain and around kz in the wavenumber domain. The wavelet power spectrum|W (z, kz)|2 represents the variance of the projection. Prior to the projection,the shear modulus µ(z, θ) is padded with zeros outside the domain [0, L] andthe wavelet ψ(z) is properly normalized. The results are shown in figure 8.9.The hatched area near the boundaries of the domain [0, L] represents the cone ofinfluence, where the spectrum is affected by the padding procedure and should beinterpreted with care [217].

Figure 8.9a shows the prior wavelet power spectrum |W (z, kz)|2 of the dynamicshear modulus µ(z, θ). This spectrum does not vary with depth. The posteriorspectrum shown in figure 8.9b reveals a clear reduction of the variance of the shearmodulus µ(z, θ), especially in the region near the surface. For large wavenumbers,the reduction of the variance disappears at a depth of about 2 m or 0.2 times thelargest wavelength in the experimental dispersion curve. This confirms that theSASW test easily resolves the variation of the shear modulus on a large spatialscale and close to the surface.

8.6 The foundation-soil transfer functions

In this section, the foundation-soil transfer functions H(ω) measured at the sitein Lincent are predicted for 1000 realizations of both the prior and the posteriorstochastic soil model. For each realization, the dynamic foundation-soil interactionproblem is solved in the same way and using the same parameter values as insection 4.4. The 95 % confidence region of the modulus of the transfer functionsH(ω) is estimated from the simulations. The convergence of the results is verifiedby means of a comparison of the confidence region calculated for the first and thesecond half of the realizations.

The foundation-soil transfer functions 191

Figures 8.10 and 8.11 show ten realizations and the 95 % confidence region ofthe modulus of the transfer functions H(ω), calculated with the prior and theposterior stochastic soil model, respectively. The results are compared with theexperimental data presented in section 1.4. The frequency range of interest extendsfrom 20 Hz to 150 Hz in the near field and from 20 Hz to 100 Hz in the far field.Outside this range, the coherence of the measured force and response vanishes andthe experimental data are unreliable. Moreover, the parameters L and M in theprior stochastic soil model have been chosen as a function of the frequency range ofinterest. The simulated results outside this range (below 20 Hz) should thereforebe interpreted with care.

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 8.10: Ten realizations (gray lines) and 95 % confidence region (gray area) ofthe modulus of the prior transfer function H(ω) compared with the experimentaldata (black line) for a source-receiver distance of (a) 4 m, (b) 8 m, (c) 16 m, and(d) 32 m. The modulus of the measured transfer function H(ω) is plotted as asolid line if the coherence function Γ(ω) exceeds a threshold value Γmin = 0.95.

The correspondence of the measured and the predicted foundation-soil transferfunctions in figure 8.11 is satisfactory in the low frequency range. In the highfrequency range, the overestimation of the transfer functions observed in section5.6 persists. This discrepancy between measured and predicted results might becaused by a too restrictive prior stochastic soil model, leading to a biased posteriorsoil model. The prior model only accounts for the uncertainty on the dynamic shear

192 Stochastic soil characterization by means of the SASW test

(a)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

4 m

(b)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

8 m

(c)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

16 m

(d)0 25 50 75 100 125 150

10−12

10−11

10−10

10−9

10−8

Frequency [Hz]

Dis

plac

emen

t [m

/N]

32 m

Figure 8.11: Ten realizations (gray lines) and 95 % confidence region (grayarea) of the modulus of the posterior transfer function H(ω) compared with theexperimental data (black line) for a source-receiver distance of (a) 4 m, (b) 8 m,(c) 16 m, and (d) 32 m. The modulus of the measured transfer function H(ω) isplotted as a solid line if the coherence function Γ(ω) exceeds a threshold valueΓmin = 0.95.

modulus, while the uncertainty on the material damping ratio is disregarded. Thesoil is also assumed to be horizontally layered, which might not be the case inreality. Another explanation of the discrepancy is non-linear behaviour of the soilunder the foundation.

Compared to the prior transfer functions, the variability of the posterior transferfunctions is smaller. Within the frequency range of interest, the strongest reductionof the variability is observed between 20 Hz and 50 Hz. In this range, the wavestravel through relatively shallow soil layers where they resolve only the large scalevariations of the shear modulus µ(z, θ). These variations are well resolved in theSASW test. As a result, the posterior uncertainty on the transfer functions H(ω)is relatively small. A single solution of the inverse problem in the SASW method,obtained using a deterministic inversion procedure, can therefore be used for arobust prediction of the free field vibrations in this frequency range (provided thatthe material damping ratio of the soil is known).

At lower frequencies, the wavelength of the waves in the soil is large and the

The foundation-soil transfer functions 193

waves in the soil reach deeper layers. The properties of these layers can not bedetermined from an SASW test on account of its limited resolution in terms ofdepth. A high posterior uncertainty on the foundation-soil transfer functions istherefore expected. However, this can not be observed in figure 8.11 due to theuse of a stochastic soil model where the spatial variation of the shear modulus atdepths larger than L = 6 m is not accounted for. As a result, the present studycan not be used to determine the lower bound of the frequency range where theSASW test allows for a robust prediction of the free field vibrations. It can onlybe concluded that this bound is below the frequency range of interest here.

At higher frequencies (above 50 Hz), the reduction of the variability of theposterior transfer functions in figure 8.11 is less pronounced. In this range, thewaves in the soil are affected by the small scale variations of the shear modulusµ(z, θ). These variations are poorly resolved in the SASW test, resulting in a highposterior variability of the transfer functions. This implies that the use of a soilprofile obtained from an SASW test with a deterministic inversion procedure canlead to an inaccurate prediction of the free field vibrations in the frequency rangeabove 50 Hz.

It is clear that the frequencies mentioned in the above discussion are not general.They are only valid for SASW tests comparable to the test in Lincent, where theexperimental dispersion curve is obtained in the range from about 20 Hz to about160 Hz.

These conclusions support the observations made in subsection 1.2.3. In thissubsection, the validation of prediction models for road and rail traffic inducedvibrations has been considered. In both cases, the dynamic shear modulus of thesoil is determined by an SASW test where the experimental dispersion curve isobtained in a comparable frequency range as in the present study. The predictionsare compared with experimental data, revealing a very close correspondence forroad traffic induced vibrations but not for rail traffic induced vibrations.

For road traffic induced vibrations, the frequency content is mainly situatedbetween 10 Hz and 30 Hz. In this range, the SASW test allows for a robustprediction of the free field vibrations.

For rail traffic induced vibrations, the frequency content is mainly situated in therange up to 100 Hz. In this range, the small scale variations of the shear modulushave an impact on the free field vibrations, resulting in uncertain predictions. Thisis a possible explanation for the lower accuracy of the predicted rail traffic inducedvibrations.

The uncertainty in the prediction of rail traffic induced vibrations can be reducedthrough a modification of the SASW test or the use of a different soil investigationtechnique with a finer spatial resolution. The methodology elaborated here allowsto assess if the resolution of such an alternative technique is sufficient or to design atest that delivers the required resolution. As an example, it is possible to determinethe optimal distance between the geophones used in the SCPT in order to minimizethe uncertainty in the prediction of rail traffic induced vibrations.

194 Stochastic soil characterization by means of the SASW test

8.7 Conclusion

In this chapter, the determination of the dynamic shear modulus of shallow soillayers from an SASW test is addressed in a probabilistic framework. A Bayesianupdating technique is used to identify an ensemble of soil profiles that fit theexperimental dispersion curve. First, a prior stochastic soil model is formulatedusing the information on the soil properties that is available before the SASW testis performed. Next, a Markov chain Monte Carlo method is applied to sample theprior stochastic soil model. For each sample, the theoretical dispersion curve iscalculated and compared with the experimental dispersion curve. The Metropolisrule is used to accept or reject a sample based on the correspondence of both curves.In this way, only the soil profiles that fit the experimental data are withheld and apopulation of profiles is obtained that follows the posterior stochastic soil model.The posterior model accounts for both the prior information and the measurementdata.

The prior and the posterior stochastic soil model are compared to assess theresolution of the SASW method. The resolution of the SASW method is limited interms of spatial scale and depth. Large scale variations of the shear modulus closeto the soil’s surface are well resolved. As the depth increases and the spatial scaledecreases, the resolution of the SASW test deteriorates, resulting in an uncertainsoil characterization.

Finally, the prediction of ground vibrations is considered. The foundation-soiltransfer functions measured in Lincent are calculated in a Monte Carlo simulation,using both the prior and the posterior stochastic soil model. The results of bothsimulations are compared in order to assess the robustness of the SASW method.In a limited frequency range, the waves in the soil are only affected by the largescale variations of the soil properties in a region close to the surface. Thesevariations are well resolved in the SASW test. Consequently, the variability ofthe transfer functions is relatively small. The SASW method can therefore beconsidered as a robust method for vibration predictions in this frequency range.In the lower frequency range, the waves in the soil reach deeper layers. Theproperties of these layers can not be determined from an SASW test due to itslimited resolution in terms of depth. As a result, the variability of the transferfunctions is larger. In the higher frequency range, the waves are affected by thesmall scale variations of the soil properties. These variations are poorly resolvedin the SASW test, also resulting in a high variability of the transfer functions. Inboth the lower and the higher frequency range, a soil investigation technique withan adapted spatial resolution is required for robust ground vibration predictions.To this end, the SASW test should be modified or a different soil investigationtechnique should be used. The resolution of such an alternative technique can beassessed and optimized with the methodology followed in the present chapter.

Chapter 9

Conclusions and

recommendations for further

research

9.1 Conclusions

Vibrations in the built environment are a matter of growing concern. In recentyears, much attention has therefore been paid to the development of predictionmodels for ground vibrations due to various sources, such as traffic, constructionworks, and industrial activities. These models are used to assess vibrationreduction measures in both existing and new situations. The models containelements related to the characterization of the source, the transmission of wavesthrough the soil, and the response of the building. The transmission of wavesthrough the soil is governed by the dynamic soil properties. Especially the dynamicshear modulus and the material damping ratio have a large influence. Theseproperties are determined by means of in situ tests or laboratory tests. Thecomplete characterization of the spatial variation of the soil properties is practicallyimpossible, however. As a result, soil models used in ground vibration predictionsare subjected to uncertainty. The uncertainty is of the epistemic type: it stemsfrom the lack of knowledge and can be reduced through (additional) measurements.

This thesis is concerned with the uncertainty on the dynamic soil properties inthe prediction of ground vibrations. The focus is primarily on the dynamic shearmodulus, but the methodology developed is also applicable to other dynamic soilproperties such as the material damping ratio. The problem is addressed in aprobabilistic framework, using a random process to model the variation of thedynamic shear modulus with depth. The variation in the horizontal direction isdisregarded.

195

196 Conclusions and recommendations for further research

A forward problem is considered, where a given (synthetic) stochastic soilmodel is used to calculate the free field vibrations due to a hammer impact ona small foundation. The shear modulus is modelled as a non-Gaussian translationprocess, characterized by a marginal probability distribution and a covariancefunction. The shear modulus is thus obtained as a memoryless transformation ofan underlying Gaussian process. This Gaussian process is discretized by meansof a Karhunen-Loeve decomposition, in a similar way as in the stochastic finiteelement method. The stochastic problem is solved by means of a Monte Carlomethod: a large number of realizations of the random shear modulus is generatedand for each realization a deterministic dynamic soil-structure interaction problemis solved. To this end, a subdomain formulation is used where the foundation ismodelled with the finite element method and the soil with the boundary elementmethod. The boundary element method is based on the Green’s functions ofthe soil. These are calculated in the frequency-wavenumber domain by meansof a direct stiffness formulation, and efficiently transformed to the frequency-space domain with Talman’s logarithmic Hankel transformation algorithm. Theoriginal algorithm is improved through the use of a window and a filter to mitigateartifacts caused by the Gibbs phenomenon. The direct stiffness formulation andthe Hankel transformation algorithm are implemented as a MATLAB toolbox(ElastoDynamics Toolbox 2.0, EDT 2.0).

Using the synthetic stochastic soil model, it is observed that the variability ofthe free field response is frequency dependent. In the low frequency range, thewavelength of the waves in the soil is large compared to the correlation length ofthe random shear modulus. The waves do not resolve the spatial variation of theshear modulus and the variability of the response is limited. In the high frequencyrange, the waves do resolve the variation of the shear modulus and the variabilityof the free field vibrations increases.

The inverse problem in the SASW method is addressed in a probabilistic way. ABayesian approach is followed to determine the uncertainty on the dynamic shearmodulus derived from the experimental dispersion curve. First, a prior stochasticsoil model is constructed. This model reflects the information on the dynamicsoil properties that is available before the SASW test. Next, a Markov chainMonte Carlo method is applied to update the prior model using the experimentaldispersion curve: random samples are drawn from the prior stochastic soil modeland withheld only if the corresponding theoretical dispersion curve is close to theexperimental curve. In this way, an ensemble of acceptable soil profiles is obtained.This ensemble represents the posterior stochastic soil model: it combines the priorknowledge and the measurement results. Inspection of the ensemble of acceptablesoil profiles reveals that relatively large scale variations of the shear modulus ina region near the surface are well resolved. The resolution of the SASW testdeteriorates as the depth increases and the spatial scale decreases.

The synthetic soil model in the forward problem is replaced with the posteriorstochastic soil model obtained from the SASW test. The free field vibrations due toa hammer impact on the foundation are recalculated. In this way, it is possible to

Recommendations for further research 197

assess the influence of resolution of the SASW method on the prediction of groundvibrations. At very low frequencies, the waves travel through deep soil layers whoseproperties are not resolved in the SASW test. As a result, the free field vibrationsare expected to be uncertain. This is not observed in the simulations, however, dueto the use of a soil model where the spatial variation of the shear modulus at largedepths is disregarded. At high frequencies, the waves in the soil are affected bythe small scale variations of the shear modulus. These variations are not resolvedin the SASW test and therefore uncertain. Consequently, the free field vibrationsare also uncertain. In the intermediate frequency range, the waves travel throughrelatively shallow soil layers and are not affected by the small scale variations ofthe shear modulus. The resulting uncertainty on the free field vibrations remainslimited. In this range, a soil profile obtained from the SASW test can be usedfor robust predictions of ground vibrations, provided that the material dampingratio is known. For the SASW test considered in the present thesis, where theexperimental dispersion curve is obtained between 20 Hz and 160 Hz, the lowerbound of this frequency range is below 20 Hz and the upper bound is about 50 Hz.

These conclusions support the observations made in the introduction of thethesis, where the validation of prediction models for road and rail traffic inducedvibrations is considered. In both cases, the dynamic shear modulus of the soil isdetermined by an SASW test where the experimental dispersion curve is obtainedin a comparable frequency range as in the present study. The predictions arecompared with experimental data, revealing a very close correspondence for roadtraffic induced vibrations but not for rail traffic induced vibrations. The frequencycontent of road traffic induced vibrations is mainly situated in the range where theSASW test allows for robust predictions. This is not true for rail traffic inducedvibrations, where the frequency content is situated at higher frequencies. Railtraffic induced vibrations are affected by the small scale variations of the soilproperties that can not be determined from the SASW test. As a consequence,the predictions are uncertain. This is a possible explanation for the lower accuracyof the predicted rail traffic induced vibrations. For more robust predictions, a soilinvestigation technique with a finer spatial resolution should be used.

9.2 Recommendations for further research

A remaining difficulty in the methodology elaborated in this thesis is the selectionof a prior stochastic soil model in the Bayesian updating scheme. Especially thechoice of a prior correlation length of the shear modulus is particularly delicate.This length determines the scale of variation of the prior and, consequently,posterior stochastic shear modulus. From a Bayesian perspective, its value reflectsthe degree of belief that the shear modulus of the soil exhibits sharp variations. Acertain level of subjectivity is therefore inherent in the method. However, furtherresearch can lead to a better founded choice of the prior correlation length.

198 Conclusions and recommendations for further research

While only the SASW test is considered in the present work, the methodology isgeneric and can also be applied to assess the resolution of other tests, such as theSCPT. Moreover, it can serve as a tool to design tests that deliver the resolutionneeded for an accurate prediction of the ground vibrations in a specific situation.As an example, the distance between the geophones used in the SCPT can beoptimized in order to minimize the uncertainty in the prediction of rail trafficinduced vibrations.

The same methodology can also be applied to determine other dynamic soilproperties, such as the material damping ratio. The material damping ratio has alarge impact on the transmission of waves through the soil and its characterizationis indispensable for a realistic estimation of the uncertainty in the prediction ofground vibrations. The primary obstacle is the lack of a well-established methodto determine the damping ratio, such as the SASW test and the SCPT for thedetermination of the shear modulus. While Rix and Lai [175] developed a methodto estimate the material damping ratio from the attenuation of surface waves,this method is much less widely used than the SASW test and the SCPT. Futureresearch should therefore focus on the development or refinement of new or existingmethods to determine the material damping ratio of soil.

In this work, the soil is assumed to be horizontally layered. The variation of thesoil properties in the horizontal direction is disregarded. In reality, this variation isgenerally expected to be less strong than the variation in the vertical direction, butit certainly has an impact on the transmission of waves through the soil. This isconfirmed by an empirical study performed at the site in Lincent: the experimentdiscussed in the present thesis is repeated but the free field vibrations are recordedat 48 locations on measurement lines in 6 different directions [13]. A considerablevariability of the free field vibrations along the different measurement lines isobserved. Future research can concentrate on the stochastic characterization ofan inhomogeneous soil with variations of the soil properties in three dimensions.The methodology elaborated in the present thesis can be used for this purpose,provided that a forward model is available to calculate the dispersion curve (or analternative measurable characteristic) of such an inhomogeneous soil. The directstiffness method is no longer applicable as it requires the soil to be horizontallylayered. The finite element method allows to model wave propagation in soilswith properties that vary arbitrarily in three dimensions, but involves a very highcomputational cost. Methods based on the enrichment of the low order polynomialapproximation of the wave field in the finite element method, such the spectralelement method [120] and the discontinuous enrichment method [232], might offera solution.

In the present thesis, the prediction of ground vibrations due to a hammer impacton a small foundation is considered. Further research can focus on the impact ofuncertain soil properties on vibrations from other sources, such as railway traffic.Due to the use of a Monte Carlo method, the existing deterministic source models[136] are directly applicable within the framework presented in this thesis. The useof a complex source model in a Monte Carlo simulation might be computationally

Recommendations for further research 199

very expensive, however. The computation time of the existing source models isdetermined by inverse Fourier transformations from the wavenumber domain tothe space domain of the coordinates parallel and orthogonal to the track. Thesetransformations are currently evaluated by means of a Filon method [72]. The useof the logarithmic Fourier transformation algorithm discussed in this thesis canlead to a more efficient source model that can be used in a Monte Carlo simulation.

Finally, the stochastic soil model in the present thesis can be coupled to astochastic building model to account for all uncertainties involved in the predictionof ground-borne vibrations and re-radiated noise in buildings. The uncertainty inthe building model stems from a large number of variables, related to structuraland non-structural parts of the building. It is very difficult or almost impossibleto model this uncertainty with the classical parametric stochastic method usedin this work, where uncertain parameters are represented by random variables orrandom processes. Methods such as statistical energy analysis offer a solutionfor predictions at very high frequencies, but these methods are not applicablein the frequency range of interest here. As an alternative, a non-parametricstochastic approach can be followed [200]. In a non-parametric approach, themodal mass, damping, and stiffness matrices of a structure are modelled as randommatrices. The probability distributions of these matrices are constructed by meansof the maximum entropy principle [107]. Given the available information on therandom matrices, this principle allows to determine the probability distributionthat maximizes the uncertainty. The available information usually consists of thealgebraic properties of the modal system matrices that have to be satisfied (such assymmetry or positive definiteness), complemented by the properties of the meansystem and a set of dispersion parameters that determine the variability of eachrandom matrix. In this way, the overall contribution of all sources of uncertaintyis represented by a minimal number of (dispersion) parameters. Due to thedirect construction of the probability distributions for the modal system matrices,the non-parametric approach allows to account for both data uncertainties andmodel uncertainties. This is an advantage over the classical parametric approach,where model uncertainties are usually disregarded. The disadvantage of the non-parametric approach is the vague physical meaning of the dispersion parametersdetermining the variability of the modal system matrices. The non-parametricapproach has already been used in the field of dynamic soil-structure interaction tostudy the uncertainty on the transfer functions from a tunnel to a nearby building[12]. In that study, however, only the uncertainty in the building model has beenaccounted for. The combination with the stochastic soil model elaborated in thepresent thesis allows to account for the uncertainty in the soil model as well.

200 Conclusions and recommendations for further research

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Curriculum vitae

Mattias Schevenels 12 July 1979, Sint-Truiden (Belgium)

Education

2002–2007

PhD student, Department of Civil Engineering, K.U.Leuven

1997–2002

MSc in Engineering - Architect (Burg. ir.-architect), K.U.Leuven

1993–1997

High school, Mathematics-Sciences, Sint-Gertrudisinstituut Landen

Work

2002–2007

Research assistant, Department of Civil Engineering, K.U.Leuven

Publications

International journal papers

M. Schevenels, G. Lombaert, G. Degrande, and D. Clouteau. The wavepropagation in a beam on a random elastic foundation. Probabilistic EngineeringMechanics, 22:150–158, 2007.

M. Schevenels, G. Lombaert, G. Degrande, D. Degrauwe, and B. Schoors. TheGreen’s functions of a vertically inhomogeneous soil with a random dynamic shearmodulus. Probabilistic Engineering Mechanics, 22(1):100–111, 2007.

221

222 Curriculum vitae

G. Degrande, M. Schevenels, P. Chatterjee, W. Van de Velde, P. Holscher,V. Hopman, A. Wang, and N. Dadkah. Vibrations due to a test train at variablespeeds in a deep bored tunnel embedded in London clay. Journal of Sound andVibration, 293(3-5):626–644, 2006. Proceedings of the 8th International Workshopon Railway Noise, Buxton, UK, 8-11 September 2004.

M. Schevenels, G. Degrande, and G. Lombaert. The influence of the depth ofthe ground water table on free field road traffic induced vibrations. InternationalJournal for Numerical and Analytical Methods in Geomechanics, 28(5):395–419,2004.

Conference papers

M.F.M. Hussein, L. Rikse, S. Gupta, H.E.M. Hunt, G. Degrande, J.P. Talbot,S. Francois, and M. Schevenels. Using the PiP model for fast calculation ofvibration from a railway tunnel in a multi-layered half-space. In 9th InternationalWorkshop on Railway Noise, Munich, Germany, September 2007. Accepted forpublication.

M. Schevenels, G. Lombaert, Degrande G., and S. Francois. Stochasticsoil characterization by means of the SASW test for the prediction of groundvibrations. In 9th US National Conference on Computational Mechanics, SanFrancisco, California, USA, July 2007. Accepted for publication.

M. Schevenels, G. Lombaert, Degrande G., and S. Francois. A probabilisticassessment of resolution in the SASW test and its impact on the prediction ofground vibrations. In 7eme Colloque National AFPS, Paris, France, July 2007.Accepted for publication.

M. Arnst, R. Cottereau, Q.A. Ta, R. Taherzadeh, D. Clouteau, G. Lombaert,M. Schevenels, G. Degrande, and M. Bonnet. Etude experimentale de la variabilitespatiale du comportement dynamique d’un sol heterogene. In 7eme ColloqueNational AFPS, Paris, France, July 2007. Accepted for publication.

S.A. Badsar, G. Degrande, M. Schevenels, and G. Lombaert. Application of theCLM method for the solution of the inverse problem in the SASW method. In 4thInternational Conference on Earthquake Geotechnical Engineering, Thessaloniki,Greece, June 2007. Accepted for publication.

M. Schevenels, G. Lombaert, G. Degrande, and S. Francois. The characterizationof the dynamic soil properties by means of a Monte Carlo inversion of theRayleigh wave dispersion curve. In 4th International Conference on EarthquakeGeotechnical Engineering, Thessaloniki, Greece, June 2007. Accepted forpublication.

Curriculum vitae 223

G. Lombaert, G. Degrande, M. Schevenels, and S. Francois. Vibrations due tohigh speed trains: numerical prediction and results from in situ measurements.In Computational Methods in Structural Dynamics and Earthquake Engineering -COMPDYN 2007, Rethymno, Crete, Greece, June 2007. Accepted for publication.

M. Schevenels, G. Lombaert, G. Degrande, and S. Francois. A probabilisticassessment of resolution in the SASW test and its impact on the predictionof ground vibrations. In Computational Methods in Structural Dynamics andEarthquake Engineering - COMPDYN 2007, Rethymno, Crete, Greece, June 2007.Accepted for publication.

M. Schevenels, G. Lombaert, G. Degrande, M. Arnst, and S. Francois. Thecharacterization of the dynamic soil properties by means of a Monte Carlo inversionof the dispersion curve. In 7th World Congress on Computational Mechanics, LosAngeles, California, USA, July 2006.

M. Schevenels, G. Lombaert, G. Degrande, G. Degrauwe, and B. Schoors. Thewave propagation in a vertically inhomogeneous soil with a random dynamic shearmodulus. In C.A. Mota Soares, editor, Proceedings of the 3rd European Conferenceon Computational Mechanics, Lisbon, Portugal, June 2006.

M. Schevenels, G. Lombaert, and G. Degrande. The Green’s functions of asoil with a random dynamic shear modulus. In Proceedings of the 7th NationalCongress on Theoretical and Applied Mechanics, Mons, Belgium, May 2006.

M. Schevenels, G. Lombaert, and G. Degrande. The stochastic finite elementmethod using the non-Gaussian Karhunen-Loeve decomposition. In C. Soize,editor, Proceedings of the 6th European Conference on Structural Dynamics:Eurodyn 2005, Paris, France, September 2005.

M. Schevenels, G. Lombaert, G. Degrande, D. Degrauwe, and B. Schoors. TheGreen’s function of a layered soil with non-Gaussian characteristics. In J.L.Bento Coelho, editor, Proceedings of the 12th International Congress on Soundand Vibration, Lisbon, Portugal, July 2005.

G. Lombaert, M. Schevenels, G. Degrande, and D. Clouteau. The wavepropagation in a beam on a random elastic foundation. In G. Augusti,G.I. Schueller, and M. Ciampoli, editors, Proceedings of the 9th InternationalConference On Structural Safety And Reliability: ICOSSAR 2005, pages 2419–2426, Rome, Italy, June 2005.

M. Schevenels, G. Lombaert, and G. Degrande. Application of the stochasticfinite element method for Gaussian and non-Gaussian systems. In ISMA2004International Conference on Noise and Vibration Engineering, Leuven, Belgium,September 2004.

224 Curriculum vitae

M. Schevenels, G. Lombaert, and G. Degrande. The influence of the depth of theground water table on free field road traffic induced vibrations. In Proceedings ofthe 6th National Congress on Theoretical and Applied Mechanics, Ghent, Belgium,May 2003.

Internal reports

M. Schevenels and G. Degrande. EDT: Elastodynamics Toolbox for MATLAB.Version 2.0. Technical Report BWM-2007-07, Department of Civil Engineering,K.U.Leuven, May 2007.

S. Francois, M. Schevenels, and G. Degrande. Applicabiliby of the MASW methodon dykes. Report BWM-2007-06, Department of Civil Engineering, K.U.Leuven,March 2007.

S. Gupta, M. Schevenels, and G. Degrande. Site vibration evaluation on theArenberg campus for the planning of a hotel laboratory for nanotechnology.Report BWM-2007-05, Department of Civil Engineering, K.U.Leuven, March2007.

G. Degrande, J. Carmeliet, B. Sluys, J. Vantomme, W. Haegeman, H.R. Masoumi,S. Francois, S. Gupta, H. Janssen, P. Moonen, H. Derluyn, S. Mertens, C. Karg,and M. Schevenels. Structural damage due to dynamic excitations: a multi-disciplinary approach. Third annual report BWM-2007-04, Department of CivilEngineering, K.U.Leuven, January 2007.

M. Arnst, Q.A. Ta, R. Taherzadeh, R. Cottereau, M. Schevenels, G. Lombaert,D. Clouteau, M. Bonnet, and G. Degrande. Measurements at a site inLincent: transfer functions, dispersion curves and seismograms. Technical report,Laboratoire de Mecanique des Sols, Structures et Materiaux, Ecole Centrale deParis, 2006.

G. Lombaert, M. Schevenels, and G. Degrande. Trillingsmetingen in een woningaan de Ieperseweg 76 te Roeselare. Technical Report BWM-2006-05, Departmentof Civil Engineering, K.U.Leuven, March 2006.

M. Schevenels, G. Lombaert, G. Degrande, and M. Arnst. Measurement andprediction of the soil’s transfer function at a site in Lincent. Technical ReportBWM-2006-03, Department of Civil Engineering, K.U.Leuven, February 2006.

M. Schevenels. SIGFUN: a MATLAB toolbox for signal processing in civilengineering. Technical Report BWM-2006-02, Department of Civil Engineering,K.U.Leuven, February 2006.

Curriculum vitae 225

M. Schevenels. Stochastic finite elements with Gaussian characteristics. TechnicalReport BWM-2004-09, Department of Civil Engineering, K.U.Leuven, November2004.

M. Schevenels, G. Degrande, G. Lombaert, and D. Dooms. Trillingsmetingen ineen woning aan de Loverstraat 37 te Sint-Baafs-Vijve. BWM-2004-05, Departmentof Civil Engineering, K.U.Leuven, May 2004.

G. Degrande, J. Maeck, G. Lombaert, S. Jacobs, and M. Schevenels. Trillingen tengevolge van metroverkeer in de Briefdragerstraat te Sint-Jans-Molenbeek. ReportBWM-2003-15, Department of Civil Engineering, K.U.Leuven, October 2003.

M. Schevenels, G. Degrande, and S. Jacobs. Vibration measurements at the site ofthe Flemish Administrative Centre in Leuven. Report BWM-2003-14, Departmentof Civil Engineering, K.U.Leuven, September 2003.

G. Lombaert, M. Schevenels, and G. Degrande. Trillingsmetingen in het vrijeveld naast de spoorlijn Brussel-Denderleeuw in de Peter Benoitstraat te Groot-Bijgaarden. Report BWM-2002-12, Department of Civil Engineering, K.U.Leuven,December 2002.