Computational Methods in Stochastic Dynamics

345 Pages·2011·10.422 MB·English

by  Gerhart I. Schuëller (auth.), Manolis Papadrakakis, George Stefanou, Vissarion Papadopoulos (eds.)

Computational Methods in Stochastic Dynamics
Computational Methods in Applied Sciences
Volume22
SeriesEditor
E.On˜ate
InternationalCenterforNumericalMethodsinEngineering(CIMNE)
TechnicalUniversityofCatalunya(UPC)
EdificioC-1,CampusNorteUPC
GranCapita´n,s/n
08034Barcelona,Spain
[e-mail]
www.cimne.com
Forothertitlespublishedinthisseries,goto
www.springer.com/series/6899
Manolis Papadrakakis George Stefanou
(cid:2)
Vissarion Papadopoulos
Editors
Computational Methods
in Stochastic Dynamics
ABC
Editors
ManolisPapadrakakis VissarionPapadopoulos
NationalTechnicalUniversityofAthens NationalTechnicalUniversityofAthens
InstituteofStructuralAnalysisand InstituteofStructuralAnalysisand
SeismicResearch SeismicResearch
9IroonPolytechniouStr. 9IroonPolytechniouStr.
ZografouCampus ZografouCampus
15780Athens 15780Athens
Greece Greece
[e-mail] [e-mail]
GeorgeStefanou
NationalTechnicalUniversityofAthens
InstituteofStructuralAnalysisand
SeismicResearch
9IroonPolytechniouStr.
ZografouCampus
15780Athens
Greece
[e-mail]
ISBN978-90-481-9986-0 e-ISBN978-90-481-9987-7
DOI10.1007/978-90-481-9987-7
SpringerDordrechtHeidelbergLondonNewYork
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Preface
At the dawn of the twenty-first century, the considerable influence of inherent
uncertainties on structural behavior has led the engineering community to recog-
nizethe importanceof a stochasticapproachtostructuralproblems.Issuesrelated
to uncertainty quantification and its influence on the reliability of the computa-
tionalmodels,arecontinuouslygaininginsignificance.Inparticular,theproblems
ofdynamicresponseanalysisandreliabilityassessmentofstructureswithuncertain
system andexcitationparametershavebeenthesubjectofextensiveresearchover
thelasttwodecadesasaresultoftheincreasingavailabilityofpowerfulcomputing
resourcesand technology.This bookfocuseson advancedcomputationalmethods
andsoftwaretoolswhichcanhighlyassistintacklingcomplexproblemsinstochas-
tic dynamic/seismic analysis and design of structures. The selected chapters are
authoredbysomeofthemostactivescholarsintheirrespectiveareasandrepresent
someofthemostrecentdevelopmentsinthisfield.
Thiseditedbookisprimarilyintendedforresearchersandpost-graduatestudents
who are familiar with the fundamentalsand wish to study or to advance the state
oftheartonaparticulartopicinthefieldofcomputationalstochasticstructuraldy-
namics.Nevertheless,practicingengineerscouldbenefitaswellfromitasmostcode
provisionstend to incorporateprobabilisticconceptsin the analysisand design of
structures.Thebookconsistsof16chapterswhichareextendedversionsofpapers
presented at the recent COMPDYN 2009 and SEECCM 2009 Conferences. The
chapters can be grouped into several thematic topics including dynamic response
variabilityandreliabilityofstochasticsystems,riskassessment,stochasticsimula-
tion of earthquake ground motions, efficient solvers for the analysis of stochastic
systems,dynamicstabilityandstochasticmodelingofheterogeneousmaterials.
InChapter1,G.I.Schue¨llergivesanoverviewofthewellestablisheddetermin-
isticmodelreductiontechniques(Guyanreduction,componentmodesynthesis)and
approachesforefficientuncertaintypropagationinstructuraldynamics.Therecent
advancesinthecombinationofthesetwofieldsarereviewedandmethodologiesfor
theconsiderationofuncertaintywhenusingmodelreductionschemesarepresented.
ThecapabilitiesofaKarhunen-Loe`veexpansionofthesubstructurematricesandof
a random combination of substructures are investigated. It is shown that the pro-
posed approaches can combine high accuracy with a substantial reduction of the
computationalcostcomparedtoMonteCarlosimulation(MCS)ofthefullmodel.
v
vi Preface
InChapter 2, S. Krenkand J. Høgsbergpresenta design principleforresonant
controlof a SDOF system by second order filters. The basic idea is the introduc-
tion of a resonantforce with frequencytuning that results in splitting the original
resonantmodeinto two modeswith equaldampingratio.The designprocedureis
developedforresonantdisplacementandaccelerationfeedback,respectively,based
onacombinationof“equalmodaldamping”andapproximatelyequalresponseam-
plitudesofthetwomodes.Theprocedureisextendedtocollocatedresonantcontrol
ofthelowermodesofMDOFsystems.Anexplicitdesignapproachforthecontrol
parameters of MDOF systems is developed that includes the effect of the higher
modesviaaquasi-staticapproximation.Theefficiencyofresonantdampingisillus-
tratedbyapplicationtoabenchmarkexampleforstochasticwindloadonahigh-rise
building.
S.K. Au presents a novel approach for improving importance sampling in the
estimationofthefirstpassageprobabilityofnonlinearhystereticsystemssubjected
tostochasticearthquakeexcitations(Chapter3).Insteadofusingfixeddesignpoint
excitations, the importance sampling distribution is constructed using a stochas-
tic process, called “adapted process”. A stochastic controlalgorithm is developed
for the adapted process where the objective function reflects the expected energy
neededtonextyieldaswellastheexitvelocity.Thevariancereductionefficiency
of the “optimal” control law is investigated through a numerical example involv-
ingaSDOFelasto-plasticstructure.Itisshownthattheproposedapproachleadsto
substantialreductionofthe“unitCOV”oftheimportancesamplingestimatorwhile
maintainingtherequiredcomputationaleffortatareasonablelevel(comparableto
thatofMarkovChainMonteCarlo-basedapproaches).
Another approach to the solution of the first passage problem for earthquake
engineeringapplicationsisprovidedbyM.BarbatoinChapter4.Thischapterde-
scribes the representation of the complex envelope process for transient response
of linear structures and illustrates the use by the first passage problem. The Van-
marckeandmodifiedVanmarckeapproximatesolutionstothefirstpassageproblem
arecomparedwiththeclassicalPoissonapproximationandMCSresultsforanide-
alized linear elastic model of a steel building. The failure condition is expressed
in terms of inter-storey drifts outcrossing specified deterministic thresholds. The
retrofitofthebenchmarkstructurewithviscousdampersisalsoconsidered,allow-
ingtoillustratetheuseoftheclosed-formapproximationsofthefailureprobability
for non-classicallydampedlinear elastic systems. The results presentedshow that
the two Vanmarckeapproximationscan improveconsiderablythe estimates ofthe
failure probability for the first passage problem when compared with the simpler
Poissonapproximation.
A computationally efficient method for the stochastic analysis of large lin-
ear structural systems subjected to seismic actions is developed by P. Cacciola
and G. Muscolino in Chapter 5. The method is based on modal analysis for the
evaluation of stochastic response along with a modal correction approach, which
includesthecontributionoftheneglectedmodes.Furthermore,apreviouslydevel-
opedmethodfortheevaluationofspectrum-compatibleevolutionarypowerspectra
of ground motions is presented. Some numerical results show the accuracy and
Preface vii
efficiency of the proposed technique for determining the spectral moments of the
response.
InChapter6,S.M.Elachachietal.investigatetheeffectofsoilspatialvariability
(correlationlength,coefficientofvariation)onthereliabilityofburiednetworkssub-
jectedtoearthquakeloading.ThereliabilityanalysisisperformedusingaResponse
SurfaceModel(RSM) and the reliabilityindexis calculatedforserviceabilityand
ultimate limit states. The dynamic response of a buried pipe subjected to natural
ground motion records is computed taking into account a longitudinal variability
ofthepropertiesofthesoil.Itisshownthatthemagnitudeoftheinducedstresses
and thus the reliability of the pipe are mainly affected by four parameters:a soil-
structurelengthratio,asoil-structurestiffnessratio,astructure-jointstiffnessratio
(relativeflexibility)andthemagnitudeofsoilvariability(coefficientofvariation).
The problem of reliability-based optimization of structural systems under
stochastic loading is treated by H. Jensen et al. in Chapter 7. In the proposed
approach, a standard gradient-based algorithm with line search is used, allowing
to explore the space of the design variables most efficiently. Subset simulation is
adopted for the purpose of estimating the correspondingfailure probabilities. The
gradients of the failure probability functions are estimated by an approach based
onthelocalbehavioroftheperformancefunctionsthatdefinethefailuredomains.
Numerical results show that only a moderate number of reliability estimates has
to be performed during the entire design process. A numerical example with de-
terministicsystempropertiesisprovidedtoshowtheeffectivenessoftheproposed
approach.
Meta-modelsareapowerfultoolforthesolutionofcomplexrealisticproblemsin
stochasticdynamics.L.Pichleretal.presentamode-basedmeta-modelforthereli-
abilityassessmentoflineardynamicalsystems(Chapter8).Thereliabilityanalysis
is carried out throughthe application of the meta-modelfor estimating the modal
propertieswhich are then subsequently required to perform a mode-superposition
analysis. The proposedapproachis provedto be accurate andcomputationallyef-
ficientinbothcasesofdeterministicandstochasticloading.Theverificationofthe
resultsforthe reliabilityassessmentcarriedoutwith themeta-modelisperformed
withafullfiniteelement(FE)analysisandshowsagoodagreementoftheresponse
overthewholedurationofthestochasticgroundacceleration.
Theeffectofuncertainsystempropertiesonstructuralresponseandreliabilityis
examinedby G. Stefanou and M. Fragiadakisin Chapter 9. An efficientapproach
combiningMCSwithtranslationprocesstheoryispresentedforassessingthenon-
linearstochasticresponseandreliabilityofasteelmoment-resistingframesubjected
to transient seismic actions. The structure is modeled with a mixed fiber-based
beam-columnelement,whosekinematicsisbasedonthenaturalmodemethod.The
adopted formulationleads to the reduction of the computationalcost required for
thecalculationoftheelementstiffnessmatrix,whileincreasedaccuracycompared
to traditional displacement-based elements is achieved. The uncertain parameters
of the problemare the Youngmodulusand the yield stress, both describedby ho-
mogeneous non-Gaussian translation stochastic fields. Under the assumption of a
pre-specifiedpower spectral density functionof the stochastic fields, a parametric
viii Preface
investigationiscarriedoutprovidingusefulconclusionsregardingtheinfluenceof
the correlationlength ofthe stochastic fieldson the responsevariabilityand relia-
bilityoftheframe.
Although reliability analysis methods have matured in recent years, the num-
ber of available techniques to evaluate the risk of complex structural systems is
limited. In Chapter 10, F. Petrini et al. investigate the problem of Aeolian risk
assessment of slenderstructuresin the frameworkof performance-basedwind en-
gineering.A classification ofthe mainsourcesof uncertaintyisattemptedandthe
importanceofsomeofthem(epistemicuncertaintyoftheaerodynamiccoefficients,
modeluncertaintyofaeroelastic forces,influenceofmodeluncertaintyonrisk as-
sessment) is examinedwith referenceto an examplecase, a long span suspension
bridge.
The concept of evolutionary power spectrum is important in the context of
stochastic simulationof earthquakegroundmotions.Chapter 11by D. Schillinger
and V. Papadopoulospresentsa novelapproachcalled “methodof separation”for
theaccurateestimationofevolutionarypowerspectraofnon-homogeneousstrongly
narrow-bandstochasticfieldsencounteredinvariousengineeringapplications.The
method is developed for power spectra separable in space-frequencybut can also
efficientlytreataclassofnon-separablespectra.Themethodofseparationisbased
onthegeometricalsimilarityoffunctions.Itleadstotheestimationofevolutionary
powerspectrafromaseriesofsamplesbasedonlyonsamplemeansquareandon
anestimateofthehomogeneousFourierpowerspectrum.Thelimitationsofexist-
ingmethods(short-timeFourier,waveletandWigner-Villetransforms)areavoided
and a very good localization in both space and frequency is achieved. Numerical
examplesincludinganalyticalbenchmarkspectraaswellasspectracorresponding
tomeasuredgeometricimperfectionsofI-sectionbeamsillustratethesuperiorityof
theproposedapproachincomparisontootherestablishedtechniques.
Anenergy-basedenvelopefunctionisproposedinChapter12byS.Sgobbaetal.
enablingtocharacterizethetime-varyingamplitudeofstronggroundmotionandto
generaterealisticstochasticaccelerogramscorrespondingtoagivenearthquakesce-
nario.ThisenvelopefunctionisdirectlyrelatedtotheAriasintensityoftheground
motionandhasafunctionalformsimilartothatofalognormalprobabilitydensity
function.Inconjunctionwithsuitablepeakfactors,theenvelopefunctionmayalso
be used to predict the distribution of PGA values correspondingto a given earth-
quakescenario.
The development of robust and efficient solution algorithms is crucial for the
treatmentoflarge-scalerealisticproblemsinvolvinguncertainties.Anovelnumer-
ical algorithm for the solution of stochastic partial differentialequations(SPDEs)
arising in the context of stochastic mechanics is presented by H. Matthies and
E.ZanderinChapter13.Thealgorithmexploitsthenaturaltensorproductstructure
betweenbasisvectorsdescribingthephysical/deterministicbehaviorandabasisde-
scribingthestochasticresponseofthesystem.Asparserepresentationofinputand
outputis achieved using singular value decomposition(SVD). This sparse format
is preservedthroughoutthe computationand is also used to reduce the amountof
computation.Anumericalexampleinvolvingthe1Ddiffusionequationispresented
Preface ix
toillustratetheefficiencyoftheapproach.Itisshownthatforasufficientlyaccurate
representation of the solution, the required storage is reduced by 90% and by
50–80%asintermediatestorageisconcerned.
The problem of dynamic stability is treated in Chapter 14. J. Na´prstek and
C.Fischerinvestigatethedynamicstabilitybehaviorofanon-linearauto-parametric
system with three degreesof freedom (used as a simple mathematical model of a
slender structure on elastic subsoil exposed to strong vertical excitation). Certain
intervals of the excitation frequency are identified where the semi-trivial solution
losesitsdynamicstability andthestronghorizontalresponsecomponentsbecome
important.Post-criticalstatesofbothdeterministicandchaotictypearethoroughly
analyzedwithrespecttoexcitationfrequencyandamplitude.Itisobservedthatthe
response amplitudescan remain in acceptable limits when adequate excitation(in
termsoffrequencyandamplitude)actsthroughoutashorttimeinterval.Itismostly
not the case when the excitation has an infinite duration and the response is ap-
proachingtothesteadystate.
The last two chapters of the book are devoted to the rapidly evolving area
of stochastic modeling of heterogeneous materials. A. Ibrahimbegovic et al.
(Chapter 15) discuss the failure analysis of civil engineering structures built of
heterogeneousmaterials.ThestructuredFEmeshapproach(consistingofconstant
stresstriangularelementthatcancontaintwodifferentphasesandphaseinterface)
is proved to be much more efficient than the non-structured mesh representation
of materialheterogeneitiesfor the case of 2D two phase materials. This featureis
of specialinterestforprobabilisticanalysis, wherea largeamountof computation
is needed in order to providethe correspondingstatistics. One such case of prob-
abilistic failure analysis is also considered in this chapter, where the geometry of
thephaseinterfaceremainsuncertainsinceitisobtainedastheresultoftheGibbs
randomprocess.Thiscomputationisfurtherusedtoprovidetheappropriateproba-
bilisticdescriptionofmaterialparametersofphenomenologicalmodeloflocalized
failure in terms of correlated random fields. Subsequent Monte Carlo computa-
tionsoffailurephenomenainsimpletensiontestperformedwithsuchprobabilistic
phenomenologicalmodelclearly show the capabilityof the presentedapproachto
recoverthesizeeffectsanywherewithinarangebetweenthetwoclassicalbounds
whichareContinuumDamageMechanicsandLinearFractureMechanics.
The book closes with a stochastic mechanics approach by D. Schwarzer and
C.Proppeforthecalculationofthestatisticsofeigenfrequenciesinheterogeneous
beamsmadeofmetalfoam(Chapter16).Thespatialvariationofthelinearelastic
material properties of metal foam is quantified using random fields generated by
thespectralrepresentationandKarhunen-Loe`veexpansionmethods.Theprobabil-
ity distribution andthe correlationstructure of the randomfieldsare derivedfrom
data.MCSofeigenvibrationproblemsofbeamsofdifferentlengthareperformed.
Thecomputedeigenfrequenciesarecomparedtoexperimentaldata.Fromthiscom-
parison, it can be concluded that the heterogeneity of the material microstructure
affects the macroscopic properties. The mean values of the eigenfrequencies are
smaller than those of beamswith homogeneousmaterial of similar mean material
x Preface
properties.A scatter of theeigenfrequenciesis also observed,a factthathasto be
takenintoaccountwhenusingpartsconsistingofmetalfoam.
Thebookeditorswouldliketoexpresstheirdeepgratitudetoallcontributorsfor
theiractiveparticipationintheCOMPDYN2009andSEECCM2009Conferences
andforthe time andeffortdevotedto the completionoftheir contributionsto this
volume.Specialthanksarealsoduetothereviewersfortheirconstructivecomments
andsuggestionswhichenhancedthequalityofthebook.Finally,theeditorswould
like to thank the personnelof Springer for their most valuable supportduring the
publicationprocess.Itishopedthatthisbookwillconveythereadertotheexcite-
ment,advancesandchallengesthe computationalstochastic dynamicscommunity
facesinthenearfuture.
June2010 ManolisPapadrakakis
GeorgeStefanou
VissarionPapadopoulos