Prioritization of Code Development Efforts in Partitioned Analysis

18 Pages·2016·0.6 MB·English

Computer-AidedCivilandInfrastructureEngineering 28 (2013) 289–306
Prioritization of Code Development Efforts
in Partitioned Analysis
Joshua Hegenderfer & Sez Atamturktur∗
GlennDepartmentofCivilEngineering,ClemsonUniversity,Clemson,SC,USA
Abstract: In partitioned analysis of systems that are To guide the model validation efforts, many frame-
driven by the interaction of functionally distinct but works, such as those presented in Unal et al. (2011),
strongly coupled constituents, the predictive accuracy of Jung (2011), Bayarri et al. (2007), and Jiang and
thesimulationhingesontheaccuracyofindividualcon- Mahadevan (2007), have been developed to rigorously
stituent models. Potential improvement in the predictive and quantitatively assess model biases and uncertain-
accuracy of the simulation that can be gained through ties(Figure1).Eachoftheseframeworksdemonstrates
improving a constituent model depends not only on the that model validation efforts can be improved through
relativeimportance,butalsoontheinherentuncertainty either additional experimentation or further code de-
andinaccuracyofthatparticularconstituent.Aneedex- velopment. If additional experimentation is necessary,
istsforprioritizationofcodedevelopmenteffortstocost- the efficient and optimal selection of experimental set-
effectivelyallocateavailableresourcestotheconstituents tingscanbeachievedthroughalgorithmssuchasthose
thatrequireimprovementthemost.Thisarticleproposes presentedinJiangandMahadevan(2007)andWilliams
a novel and quantitative code prioritization index to ac- etal.(2011).However,ifcodedevelopment isselected
complishsuchataskanddemonstratesitsapplicationon as the next step in model validation, the particular el-
acasestudyofasteelframewithsemirigidconnections. emental components of the numerical model requiring
Findingsshowthatashigh-fidelityconstituentmodelsare more sophisticated modeling of physics or engineering
integrated, the predictive ability of model-based simula- principles remains unknown. Naturally, unlimited time
tionisimproved;however,therateofimprovementisde- and resources would render the prioritization of such
pendentuponthesequenceinwhichtheconstituentsare code development efforts irrelevant. However, given
improved. theinevitablelimitationsinresources,majormodelde-
ficiencies must be pinpointed and code development
1 INTRODUCTION efforts must be focused to achieve the greatest reduc-
tioninmodeluncertaintyandbias.Thescientificprob-
Numerical simulation has become a viable tool for lemofcodeprioritizationhasbeenstudiedtooptimize
investigating complex physical systems and processes softwarefaultdetection(Korel,2009)andparallelpro-
that are encountered in many civil engineering disci- cessing of finite element (FE) computer codes (Zeyao
plines,especiallywhensolelyexperiment-based,empir- andLianxiang,2004);however,quantitativeapproaches
ical studies are infeasible. Although numerical models forcodeprioritizationofcomplexnumericalmodelsare
offerversatilityinsimulatingvariousoperationalcondi- lackinginthepublishedliterature.
tions or design scenarios, they can only provide an ap- Such need becomes especially amplified for com-
proximationofreality,andthus,thereisaneedtocheck plex, heterogonous systems that are driven by the in-
the validity of model solutions against experimental teraction of functionally distinct but strongly coupled
observations. Numerical models therefore, come with constituents commonly tackled by partitioned analy-
the burden of quantifying the uncertainties and biases sis procedures. Our focus is therefore on models that
in model predictions through tasks that fall under the comprise multiple, isolated, components (or substruc-
broadconceptofmodelvalidation. tures), herein referred to as constituents. These con-
stituents interact to form a collective system, herein
referred to as coupled model. Such focus is relevant
∗To whom correspondence should be addressed. E-mail:
[email protected]. as many numerical models in civil and infrastructure
C 2012Computer-AidedCivilandInfrastructureEngineering.
⃝
DOI:10.1111/j.1467-8667.2012.00799.x
290 Hegenderfer&Atamturktur
(empirical or arbitrary parameters used in place of de-
tailed physics modeling) to represent these inherently
START
semirigid connections. This initial FE model can then
be improved by refining the definition of these knobs,
suchasthroughthedevelopmentofhigh-fidelitythree-
Baseline Models and
Uncertainties with Current dimensional,nonlinearFEmodels(constituents).Three
Experimental Design s nt possibleconstituentconnectionmodelsareidentifiedto
w ment depme be coupled to the initial simplified frame FE model.
Neeri Coelo Thesethreeconstituentsarerankedusingtheproposed
p v CPI that combines knowledge level, importance, and
Model Validation and x e
E D error contribution. The frame model is then improved
Uncertainty Quantification
incrementally by coupling the three high-fidelity con-
OR
nection models in the sequence selected by CPI, and
No
theresultingimprovementinpredictiveaccuracyofthe
Path Selection
coupledmodelisquantified.
Va lidation Algorithm
Criterion The article is organized as follows: Section 2
Met? overviews the pertinent literature on coupling algo-
Validated rithmsandrankingproceduresthatareusedasthefoun-
Model and END dation for the CPI described in Section 3. Section 4
Uncertainties
Yes Quantified introducesthecasestudyapplication,inwhichthepre-
dictions of the initially inexact FE model of the steel
Fig.1. Genericmodelvalidationframework. frame are compared against experimentally obtained
static and dynamic characteristics. In Section 5, the
CPI is deployed to rank model constituents, which are
engineering are indeed coupled models that are an developed, verified, and validated in Section 6. The
amalgam of multiple constituents or systems of con- coupling procedure used to integrate the constituent
stituents; see, for instance, the published literature on models to the frame model is described in Section 7.
soil–structure interaction (Provenzano, 2003; Qian and Section8presentsacomparisonoftheinitialFEmodel
Zhang, 1993), fluid–structure interaction (Kutay and as well as its improved variants against experimental
Aydilek,2009;Caracogliaetal.,2009),human–structure data.FinallyinSection9,concludingremarksaremade,
interaction (Macdonald, 2009; Wang et al., 2011), and limitations of the proposed approach are summarized,
thebroadfieldofsubstructuring(orsubdomains),which andthefuturedirectionisoverviewed.
essentially focuses on structure–structure interaction
(Mahjoubi et al., 2009; Mahjoubi et al., 2011). While
2 BACKGROUND
aimingtoimprovethepredictiveaccuracyofsuchcou-
pledsimulationmodels,oneobviousquestionarises:to
2.1 Phenomenonidentificationandrankingtable
achievethegreatestreductioninmodeluncertaintyand
bias,whichconstituentmustbegiventhehighestprior- Originally proposed as part of the U.S. Nuclear Re-
ityforfurthercodedevelopment? actor Commission’s (NRC) Code Scaling, Applicabil-
This article presents a code prioritization approach ity, and Uncertainty (CSAU) evaluation procedure
for coupled numerical models through the use and ex- (Boyack et al., 1989), the Phenomenon Identification
tension of a well-established ranking system, in which and Ranking Table (PIRT) is currently being used by
code development efforts are guided to effectively im- theU.S.NRCatthestartofnewprogramstorankcon-
prove the predictive capability of the coupled model. stituent phenomena from the perspective of resource
A novel, quantitative code prioritization index (CPI) allocation (ARRIA, 2003; Olivier and Nowlen, 2008;
isproposedanddemonstratedusingaproof-of-concept Tregoning et al., 2009). The purpose of PIRT is to ef-
example, in which the structural system consists of a fectively gather expert opinion about the importance
two-storysteelframebuiltinthelaboratorywithsemi- andknowledgelevelofasetofphenomena(Diamond,
rigid connections. An initial, simplified FE model is 2006).ThePIRTprocess,typicallycompletedbyacom-
developed for the frame system using beam elements, mittee of multidisciplinary experts, provides a system-
wherethebeam-to-columnconnectionscanbemodeled atic, structured, and hierarchical methodology to rank
as either fixed or pinned connections, neither of which phenomena of interest for resource allocation prob-
are representative of reality. One approach commonly lems (Boyack, 2009). The two major scoring compo-
utilized in the literature entails adding fictitious knobs nentsofPIRT,(1)theimportanceofaphenomenaand
Effortsinpartitionedanalysis 291
(2) the level of current knowledge, constitute the par-
ticularphenomena’ssensitivityorimpactontheevalu-
ation metric and the current lack of knowledge about
thephenomena(and/orparametersassociatedwiththat
phenomena), respectively. Although historically PIRT
has mainly been used in the nuclear reactor and safety
fields,thegeneralandhigh-levelapproachofthePIRT
processmakesithighlyadaptabletomanydifferentar-
easofscienceandengineering.
2.2 Partitionedanalysisprocedures
Fig.2. Couplingalgorithms:(a)Block-Jacobi,(b)Block
In partitioned analysis procedures, constituent models Gauss-Seidel,(c)BlockNewton,and(d)optimization-based
are viewed as discrete entities with data transferred at coupling.
the interface between the individual constituent codes
strong coupling algorithms are proposed in the litera-
throughcouplingalgorithms(RugonyiandBathe,2001;
ture(Figures2a–d):theBlock-Jacobimethod(Matthies
Xiongetal.,2011).Thistypeofanalysismostoftenre-
etal.,2006;FernandezandMoubachir,2005),theBlock
sultsinaniterativeprocedureinvolvingprediction,sub-
Gauss-Siedel method (Joosten et al., 2009; Matthies
stitution, and organization techniques (Felippa et al.,
et al., 2006), gradient-based Newton-like methods
2001; Larson et al., 2005; Larson, 2009). The advan-
(Heil, 2004; Matthies and Steindorf, 2002b; Matthies
tageofthepartitionedapproachstemsfromtheability
and Steindorf, 2003; Fernandez and Moubachir, 2005),
to exploit independent modeling strategies developed
and optimization-based methods (Farajpour and
in different domains (Leiva et al., 2010) in addition to
Atamturktur,2012).
time and space discretization that is most appropriate
Notethatinpartitionedanalysis,quantification,prop-
foreachconstituent(Kassiotisetal.,2011;Joostenetal.,
agation,andmitigationofuncertaintiesininputparam-
2009). This flexibility obtains solutions for highly com-
etersplayanimportantroleforthecompletevalidation
plex coupled problems while making efficient use of
ofcoupledmodels(AvramovaandIvanov,2010),which
well-establishedcodesandexpertknowledgeinvarious
iscurrentlyanactiveresearcharea.Thefocusofthisar-
fields (Ibrahimbegovic´ et al., 2004). Furthermore, the
ticle,however,isexclusivelyonimprovedsophistication
ability to solve complex problems through integrated
of constituent models to reduce the systematic bias in
parallelization on multiple sets of processors can make
coupled models predictions; therefore, uncertain input
thepartitionedapproachparticularlyefficient(Parkand
parametersaretreatedasdeterministicbestestimates.
Felippa,1983;AdeliandKamal,1993).
Coupling is most commonly classified according
to the nature of the mutually dependent parameters. 3 CODEPRIORITIZATIONINDEX
In weak coupling (also known as loose or explicit
coupling), the state of the constituents does not mu- ThePIRT,overviewedinSection2.1,suppliesaninher-
tually interact with each other’s inputs or outputs ently qualitative and subjective evaluation of a system
(Matthies and Steindorf, 2002a; Wang et al., 2004). In that is driven by the interaction of coupled constituent
partitioned analysis, if the state of the constituents is phenomena. As such, this section details the transfor-
affected by model outputs, the coupling interface is mation of PIRT into a quantitative approach through
referred to as strong coupling (also known as tight or theCPI.FollowinganapproachsimilartoHemezetal.
implicitcoupling)(MatthiesandSteindorf,2003;Zhang (2010), CPI is formulated considering the three fac-
andHisada,2004).Instrongcoupling,therepeatedex- tors that determine the importance a constituent has
ecutionofconstituentmodelstoachieveself-consistent on the overall predictive accuracy of a coupled simula-
state solutions becomes necessary (Ramanath, 2011). tion: (1) the sensitivity of the coupled system behavior
Input parameters in strong coupling problems are totheconstituentoutput,(2)theestimatedbiasesinthe
defined as either dependent or independent. The de- constituent output, and (3) the estimated uncertainties
pendentinputparametersarefunctionsoftheoutputof in the constituent. CPI, therefore, takes the following
another constituent; therefore, the coupling algorithm form:
mustevaluateandsubstitutetheseparametersasinput
CPI SA EA UA (1)
for the appropriate constituent. Finding the correct i =∥ i∥×∥ i∥×∥ i∥
values for these dependent variables is the main ques- where SA, EA, and UA, respectively, represent
i i i
tion to be solved in strong coupling problems. Various the importance (Sensitivity Analysis), estimated error
292 Hegenderfer&Atamturktur
(Error Analysis), and current level of knowledge (Un-
certaintyAnalysis)foragivenconstituenti,inanappro-
priatelychosennorm, .
∥•∥
3.1 Sensitivityanalysis(SA)
The SA term is defined as the measure of the variabil-
ityinthemodeloutputduetounitvariabilityinagiven
constituent. This term can be quantified using various
established screening, local or global sensitivity analy-
sistechniques;however,forcomplexproblemsthatare
nonlinear in nature and that involve a large number of
input parameters of varying uncertainty, global sensi-
tivity analysis techniques should be preferred (Cukier
et al., 1978). The global sensitivity analysis inherently
takes the range and shape of the probability distribu-
tions of input parameters into account. Moreover, in
global sensitivity analysis techniques, such as the anal- Fig.3. Computationof P:absoluteminimumdiscrepancy:
j
ysis of variance (ANOVA) (Moaveni et al., 2009; Frey Equation(6)(top)andconvergedminimumdiscrepancy:
and Patil, 2002), the sensitivity estimates for each pa- Equation(7)(bottom).
rameterarecomputedinthepresenceofuncertaintyof
allotherfactorsofinterest;thustakingthepossiblecor- theconstituentofinterest,whichisderivedinEquations
relations and dependencies between input parameters (4)–(7).
intoconsideration. AnabsoluteminimumdiscrepancyexistsifEquations
(4)and(5)givenbelowarebothsatisfied.
3.2 Erroranalysis(EA) δk δk r
j − j− < γ (4)
The EA term is a representative estimate of a con- P(δkj)− P(δkj−r) −
stituent’s error contribution to the total error of the
system in comparison to the experimental evidence. δkj+r −δkj < γ (5)
This term can be quantified using various metrics in- P(δk r) P(δk) +
cluding model form error estimates for separate ef- j+ − j
fect experiments as defined in Higdon et al. (2008), whereγ isthelimitingslopeasindicatedinFigure3and
OberkampfandBarone(2006),Rebbaetal.(2006),and r represents the number of runs over which the slope
Atamturkturetal.(2011). is computed (see Figure 3); k is the index number of
In this study, for a given constituent, EA is defined computer runs corresponding to the minimum discrep-
as a measure of the constituent “correction” necessary ancy over all simulation runs; and δkj is the minimum
suchthaterrorinthecoupledmodelpredictionsismin- computeddiscrepancyacrossallsimulationruns.Inthis
imized (given a set of uncertainty attributed to other case,thethresholdvaluefordiscrepancyisselectedac-
constituents)asshowninEquations(2)and(3) cording to Equation (6); otherwise, a converged mini-
mumdiscrepancyexistsandthethresholdvaluefordis-
Pj ={P :δj(P,α)<δ tj} (2) crepancyiscalculatedaccordingtoEquation(7);
δt δk (6)
1 n j = j
EA min P P P (3)
0 j 0
= n − δt β δS δk δk (7)
!j=1" # #$% j = j − j + j
# #
where n is the number of experiments; P0 represents where β represents a pe"rcentage$of the total allowable
the nominal value for the fictitious knob; P is the se- reduction discrepancy to limit excessive changes in the
j
lectedparametervaluefortheconstituentatthejthex- nominal value of the knob (see Figure 3) and δS rep-
j
periment;δ (P,α)isthediscrepancyoftheconstituent resents the maximum computed discrepancy across all
j
atthejthexperimentdefinedasafunctionofcollective simulationruns.
parametersoftheconstituents;andα istheuncertainty The EA term in Equation (3) seeks to determine
assignedtoallotheruncertainparameters.Theδt isthe thenecessarydeviationfromtheinitialparametervalue
j
discrepancy threshold value at the jth experiment for over all available experiments such that the relative
Effortsinpartitionedanalysis 293
difference between model predictions and experimen-
tal results (referred to herein as discrepancy) is min-
imized. To achieve this minimization, the numerical
modelparametersofinterestaresampledwithinaplau-
sible range defined for parameter of interest, P, and
a set percentage variability, α, for all other param-
eters, thus accounting for potential cross-correlations
between parameters through the α variability term.
Computer runs can be generated using various sam-
plingdesignsthatthoroughlyexplorepossiblevaluesfor
P in an efficient manner. The EA value realized from
Equation (3) is represented as a percentage of the ini-
tially assumed parameter value, P. Therefore, higher
0
EAvaluesindicateagreaterestimatederrorforthepa-
rameterofinterest.
Thevaluesofr,γ,andβ(fromEquations(4)–(7))are
assigned according to the specific application of inter-
est. If discrepancy throughout the investigated domain
is constant (the calculated slope is below the limiting
value of γ) or if the discrepancy behavior is divergent
Fig.4. Framestructurebuiltinthelaboratory(left)and
for any specific experiment j, then a meaningful and
frameconnections(right):topconnection(top),middle
confidentevaluationof P isdifficult.Thus,ifthegiven
j connection(middle),baseconnection(bottom).
experiment is not sensitive to parameter P, a selection
of P would be unfounded; therefore, the contribution
j theCPItermsischosensuchthateachtermisscaledto
fromsuchexperimentsisexcludedfromtheanalysis.
amaximumofoneoverallpossibleconstituents;there-
Notethatifmodelcalibrationofconstituentsiscom-
fore,CPIvaluesare,withoutlossofgenerality,bounded
pletedpriortotheutilizationoftheCPImetric,thecal-
betweenzeroandone,andhigherCPItermsareconsid-
ibrated parameter values for the parameters would be
eredashigherpriority.
usedastheinitialvaluesfortheEAanalysis.Whensuch
calibrationactivitiesarenotavailableapriori,thepro-
posed method gives the best possible guidance by di-
4 OVERVIEWOFTHECASESTUDY
recting the focus of the decision makers and code de-
APPLICATION
velopers to the constituent that is mostly responsible
for the inaccuracies of the coupled model prediction.
4.1 Casestudystructure:laboratoryspecimen
The inaccuracies in the selected constituent may orig-
inate from either the incompleteness of the modeled The small-scale, prototype structure is a two-story
physics or engineering principles or the imprecision in single-bay steel frame shown in Figure 4, in which all
themodelparameters. membersoftheframearemadeofA36mildsteel.The
connections are secured with two SAE J429—Grade 5
boltsoneachside,andallboltsaretorquedto9.0Nm.
3.3 Uncertaintyanalysis(UA)
Thegeometricdimensionsofthetestframearegivenin
The UA term can be quantified for a particular model Table1.
constituent according to experimental test results or
qualitativeexpertopinion.Herein,however,thecurrent
4.2 Initialnumericalmodel
levelofknowledge,thatis,uncertaintyanalysis,foreach
constituent will be treated as a binary number, where The baseline FE model for the frame (henceforth re-
zerorepresentsaphenomenonforwhichtheknowledge ferred to as Model #1) is created in ANSYS v.13.0 us-
levelismatureandwhereonerepresentsaphenomenon ing BEAM188 elements as shown in Figure 5. These
thatisnotyetwellknown.Ineffect,asrecommendedin are two-noded six-degrees-of-freedom per node linear
the PIRT construction process, the UA term restricts elements that consider the cross-section and orienta-
codeprioritizationeffortsonlytopoorlyorpartiallyun- tionofthemember(ANSYS,2010).Anisotropiclinear
derstoodphenomena. elasticconstitutivemodelisusedwithYoung’smodulus
Therefore,theCPItermismaximizedwhenSA,EA, specifiedas200GPa(29,000ksi)andPoisson’sratioas
andUAareallhigh.Herein,atypeofnormforeachof 0.33, which are typical values for mild structural steel.
294 Hegenderfer&Atamturktur
Table1
Geometricpropertiesofthelaboratoryframestructure
Steelframemembergeometricproperties
Member Length(cm) Cross-sectiontype Cross-sectiondimensions(cm)
Columns 63.5 Angle 5.08 5.08 0.32
× ×
Beam 124.46 Flat 5.08 0.32
×
Baseconnectingtabs 5.08 Angle 5.08 5.08 0.32
× ×
Columnbaseplates 1.27 Flat 15.24 15.24
×
Framebase 1.27 Flat 121.92 243.84
×
Boltproperties
Bolttype Boltdiameter(cm)
SAEJ429—Grade5 0.66
Fig.6. Staticexperimentsetup:cableandpulleysystem
(left),measurementlocations(right).
top and middle levels are pinned. After the selection
Fig.5. BaselineFEmodel(Model#1)fortheframe of a suitable connection type, Model #1 will ultimately
structure. be used as a reference while determining the improve-
ment gained in predictive accuracy by refining the way
in which beam-to-column connections are represented
Similar to the approach used by Zapico et al. (2008),
intheFEmodel.
geometric offsets are considered when modeling the
beam-to-column connections (also known as modeling
4.3 Experimentalcampaign
tocentroid).Thetwo-boltconnectionsoftheframeare
expected to exhibit semirigid behavior with unknown 4.3.1 Static testing. Static testing is conducted using a
stiffnesscharacteristics. cable and pulley system designed to apply a point load
Intheliterature,suchsemirigidconnectionsareoften in the horizontal direction as shown in Figure 6. Fed-
approximatedaseitherfixedorpinneddependingupon eral C8IS Dial Deflection indicators are positioned at
the number of bolts, the bolt pattern, and expert judg- the locations shown in Figure 6 to measure horizontal
ment(LeeandMoon,2002;Galva˜oetal.,2010).There- deflection. Six separate static tests are conducted us-
fore, an initial numerical model (Model #1) is built ing point loads with increased amplitude and applied
withsimilarapproximationscommonlyimplementedin at a different location on the structure as shown in
practice. For Model #1, to determine a suitable con- Figure 6, while displacements are measured at four
nection type, either fixed or pinned connection, model points(L –L )alongasinglecolumnforTests1–3and
1 4
predictions are compared against experimentally ob- two points (L and L ) on the beams for Tests 4–6.
5 6
tainedstaticanddynamicresponse.Inthefixedmodel, The displacement results for the static tests, shown in
all connections are assumed to be fixed. However, for Figure 7, are compared to both the pinned and fixed
stability, in the pinned model the rotational degrees of connection FE models in Table 2. The static tests are
freedom at the base connections as well as the top- completed at force levels where the material has not
andmiddle-leveltorsionaldegreesoffreedomarefixed, yielded, that is, it is still in the elastic regime as is
whileallremainingrotationaldegreesoffreedomatthe demonstratedinthelinearresponseshowninFigure7.
Effortsinpartitionedanalysis 295
Static Test 6: Measurement PointL
6
130
120
110
N)100
(
e
c 90
r
o
F 80
70
Simm.fixed
60 Simm.pinned
Exxperiment
50
0 5 10 15 20
Displacement (mm)
Fig.7. StatictestcomparisonbetweentwoalternativeFE
models(pinnedconnectionsandfixedconnections)and
Fig.8. Modeshapes:experimentaltesting(top)andFE
experimentaldata.
modelpredictions(bottom).
Table4
4.3.2 Modal testing. Model 4507B Bru¨el and Kjær MACofexperimentallycollectedmodes
(B&K)unidirectionalaccelerometersaredistributedto
Mode 1 2 3
88 locations, 6 inches apart throughout all beams and
columns(seeFigure4),andaModel8207B&Kmodal 1 1 0.003 0.002
impact hammer is used to excite the structure. Each 2 0.003 1 0.001
hammer strikeis repeated five times to reduce degrad- 3 0.002 0.001 1
ing effects of noise through averaging. Using Reflex
software from B&K, natural frequencies (see Table 3)
calculated(Table4).Thelowoff-diagonaltermsinTa-
and mode shapes (see Figure 8) are extracted from
ble 4 demonstrate the linear independence of each of
the experimental data. The rational fraction polyno-
theexperimentallydeterminedmodes.
mial parameter estimation technique (Richardson and
Formenti,1982)isimplementedtogeneratestabilitydi-
4.4 Test-analysiscorrelation
agramsformodeselection.Thefirstthreeglobalmodes
of the structure are identified as shown in Figure 8 Tables 2 and 3 show that the baseline model with
and listed in Table 3. As a means of verifying the or- fixedconnectionsunderestimatesthedeformationsand
thogonality of the experimentally collected modes, the overestimates the natural frequencies, while the base-
modal assurance criterion (MAC) (Allemang, 2003) is line model with pinned connections overestimates the
Table2
Comparisonofbaselinemodelsandexperimentaldataforstatictests
Test 1 2 3 4 5 6 Mean
FixedFEaverageerror(%) 44.4 84.8 55.6 46.8 24.3 18.9 45.8
PinnedFEaverageerror(%) 32.8 196.5 76.7 18.8 100.5 126.3 91.9
Table3
Naturalfrequenciesfromexperimentaltestingandbaselinemodels
Mode Description Testfrequency(Hz) FixedFEfrequency(Hz) Error(%) PinnedFEfrequency(Hz) Error(%)
1 Sway 23.09 27.97 21.13 16.30 29.41
2 Bending 29.58 33.21 12.27 14.01 52.64
3 Torsion 37.04 47.56 28.40 25.96 29.91
296 Hegenderfer&Atamturktur
deformations and underestimates the natural frequen- tions presented in Wu and Li, 2006) and treated as
cies. As seen, neither the fixed nor the pinned connec- constituents.
tions are suitable to represent the connections of the Similar to da Silva et al. (2008), three linear, rota-
frame of interest. However, the FE model with fixed tional springs are added at each connection of the ini-
connections more accurately predicts the static tests tial FE model (Model #1), which are used to capture
with a 46.1% lower average error than the FE model rotations of the connection in all three directions. The
withpinnedconnections.Also,theFEmodelwithfixed springs are modeled using linear, COMBIN14 (spring-
connections predicts the values of the natural frequen- damper) elements (ANSYS, 2010). As the spring con-
ciesinthecorrectorderofmodes,whilepinconnections stantstendtozero,theconnectionbehavesasapin,and
yieldanincorrectmodesequence. as the spring constants tend to infinite the connection
AlthoughtheFEmodelwithfixedconnectionsyields becomes fixed. The stiffness of the bolted connections
a closer agreement to experiments and has been se- utilizedintheframestructure,however,liesomewhere
lected as the connection type to be used in Model #1, betweenthesetwoextremeswithunknownspringcon-
itstillhasanapproximately20%averageerrorfornat- stants.Also,notethatthoughthislinearrepresentation
ural frequencies and 45% average error over all static ofsemirigidspringssuppliesanimprovementbeyonda
tests, necessitating further improvement of the model fixed or pinned connection, it fails to incorporate non-
sophistication. lineareffects.
The FE model can be further improved by develop-
ingthree-dimensional,nonlinearFEmodelsofthecon-
5 IDENTIFICATIONANDPRIORITIZATIONOF nectionsthataccountforfrictionbetweenthemembers
CONSTITUENTS andpretensioninthebolts(asitwillbediscussedinSec-
tion6).Thesehigh-fidelity,nonlinearFEmodelsofcon-
The structure of interest comprises two main con- nectionsmustbedevelopedseparatelyforthetop,mid-
stituents: the steel members and the connections. dle, and base connection; however, the limitations on
Though the material behavior of mild steel is widely projectresourcesmayinhibitmodeldevelopmentofall
studied,theboltedsemirigidsteelconnectionsbetween three connections. Thus, to efficiently utilize the avail-
beamandcolumnarehighlyuncertainandareofgreat able resources, one must prioritize among these three
interest in the design and analysis communities. Lee possible constituent models to achieve the highest im-
and Moon (2002) and CEN (1997) provide provisions provement in predictive capability. Such prioritization
forapproximateanalysisofsemirigidsteelconnections; is achieved herein using the linear FE models of the
however, many assumptions are made while establish- constituentalongwiththeCPImetricintroducedinSec-
ing these provisions, such as small deformation in the tion3.TheCPIrequiresasensitivityanalysis,anderror
connection,negligibledeformationinthebeamandcol- estimate analysis, which are discussed in the following
umn compared to the deformation in the connection, sections.
andnegligibleslipdeformation.
Several investigators acknowledged and demon-
5.1 Sensitivityanalysis
strated that traditional assumptions of connections be-
ing either fixed or pinned can lead to significant er- Herein, a global effect sensitivity analysis, ANOVA, is
rorsintheanalysis of static(Leeand Moon, 2002) and utilized. The selection of the plausible ranges, within
dynamic response (da Silva et al., 2008; Galva˜o et al., whichspringconstantsmayvary,iscriticalinANOVA;
2010; Tuu¨rker et al., 2009; Wong et al., 1995), design therefore, plausible ranges are determined by varying
and sizing of members (Lee and Moon, 2002), as well individualspringconstantsfromfreetoanalmostfixed
asdesignagainstprogressivecollapse(Liuetal.,2010). conditiononeatatime,whiletheremainingspringcon-
The findings obtained in Section 4.4 of this study also stantsareleftconstantattheirnominalvalues.See,for
support these earlier studies that neither the fixed nor instance,Figure9,wheretheresultingnaturalfrequen-
thepinnedconnectionsareacceptabletorepresentthe ciesareplottedasthestrongaxisbendingstiffnesscon-
two-bolt connection used in the steel frame studied stantsfortoplevelbeamsisvaried.Asonespringcon-
herein. Therefore, while demonstrating the application stantisvaried,thenaturalfrequenciesvarybetweenan
of the proposed CPI metric, this study demonstrates upperbound(correspondingtoafixedconnection)and
mitigation of potential errors originating from the in- a lower bound (corresponding to a pinned connection)
complete and inexact modeling of the semirigid steel in an asymptotic manner. Appropriate ranges for each
connections through a rigorous modeling approach, spring constant can then be selected to ensure a semi-
where connections are parameterized (similar to the rigid behavior. As demonstrated in Figure 9, for the
structural parameter identification for flexible connec- strong axis bending stiffness constant at the top level,
Effortsinpartitionedanalysis 297
50 60
40 50
%)
z)30 y ( 40
H c
q( an
Fre20 screp 30 TTeesstt 13
Di P
10 MMooddee 12 20 P0
1
Mode 3
P
3
0
0 2 4 6 8 10
10 10 10 10 10 105 1010
Stiffness (N-m) Base Rotational Stiffness (N-m)
Fig.9. Sensitivityofnaturalfrequenciestostrongaxis Fig.10. DiscrepancyversusParameter1andselected
bendingstiffnessconstantatthetoplevel. parametervaluesforStaticTests1and3.
therangeischosentobe1.1 101–1.1 106 Nmasin-
× × 5.2 Erroranalysis
dicatedwithdashedlines.
After selecting the ranges for each stiffness con- Theparameterofinterest,P(seeTable5forparameter
stant, the sensitivities of natural frequencies and lat- numbersandcorrespondingconstituents),isallowedto
eraldisplacementstothespringconstantsareanalyzed varyfrom1.1 101 to1.1 1010 Nm;α ischosentobe
× ×
throughANOVA.AsseeninFigure9,theanalytically equal to 10%, γ is set to 0.5, n is 4, and β is chosen to
computed modes can swap their order when the stiff- beequalto20%.Onehundredscenariosaregenerated
ness constant is varied. For proper implementation of throughLatin-Hypercubesampling.Figure10plotsdis-
ANOVA on natural frequencies, a mode-pairing algo- crepancy as a function of spring constant for Parame-
rithmisdeveloped,whichoperatesonthespatialprop- ter 1 for Static Tests 1 and 3, where the discrepancy is
ertiesofeachmodetoproperlyreorderswappedmodes computedasapercentageoftheexperimentalvaluefor
according to a reference mode sequence. Table 5 lists each experiment. The plot shows the initial parameter
therelativecontributionofthestiffnessconstantstothe value and algorithm chosen parameter values for each
overall variability in the predicted features, where the experiment,respectively.
normalized sensitivities are summed for each connec- In Figure 10, P represents the nominal value for
0
tion level. Results indicate that the most sensitive con- the knob, which in this case is the stiffness constant
stituentstomodeloutputarethemiddle,base,andtop of the base connection, and P and P represent the
1 3
connections, respectively. The torsional stiffness con- selected values, P , that minimize the discrepancy of
j
stants are insensitive to the predicted features for all Model #1 against Static Tests 1 and 3, respectively, ac-
connection levels and thus, are not included in further cording to Equations (4)–(7). P shows a case where
1
discussionandareheldatnominal(i.e.,fixed)values. an absolute minimum discrepancy exists. Here, the
Table5
SensitivityanalysiswithnormalizedR2(%)valuesforeachparameter
Naturalfrequency Maxhorizontaldisplacement
Parameternumberand
descriptionofstiffness Mode1 Mode2 Mode3 %BeamTop %BeamMid %Column & Total
Base 1 Rotational(strongaxis) 47.7 36.9 65.3 18.1 7.3 37.2 212.3 212.3
2 Torsional 0.00 0.00 0.00 0.00 0.00 0.00 0.01
Middle 3 Rotational(strongaxis) 33.2 51.8 23.1 17.1 5.9 35.2 166.3 251.6
4 Rotational(weakaxis) 2.2 0.02 0.17 0.08 82.8 0.00 85.3
5 Torsional 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Top 6 Rotational(strongaxis) 12.0 11.1 11.2 13.4 4.0 27.6 79.2 136.1
7 Rotational(weakaxis) 4.9 0.30 0.32 51.3 0.02 0.00 56.9
8 Torsional 0.00 0.00 0.00 0.00 0.00 0.00 0.00
298 Hegenderfer&Atamturktur
Table6
Erroranalysisofthethreeconstituentmodels
Descriptionof Average
Constituent stiffness EA(%) constituentEA
Base Rotational(strongaxis) 49.8 49.8
Middle Rotational(strongaxis) 51.2 59.1
Rotational(weakaxis) 66.9 Fig.11. FEconnectionmodels:topconnection(left),middle
Top Rotational(strongaxis) 54.0 61.9 connection(middle),andbaseconnection(right).
Rotational(weakaxis) 69.8
phases: first, the initial frame FE model (Model #1) is
Table7
coupled with only the middle connection model result-
CPIforthethreeconstituentmodels
ing in Model #2. Second, the base connection model is
Constituent SA EA UA CPI added to Model #2 to obtain Model #3. Finally, Model
∥ ∥ ∥ ∥ ∥ ∥
#4isconstructedbyaddingthetopconnectionmodelto
Baseconnection 0.84 0.80 1.00 0.68
Model#3.
Middleconnection 1.00 0.95 1.00 0.95
Topconnection 0.54 1.00 1.00 0.54
6 DEVELOPMENT,VERIFICATION,AND
VALIDATIONOFCONSTITUENTS
selected parameter value,P, is the parameter value at
1
the computed minimum discrepancy. P shows a case
3
6.1 Constituentmodeldevelopment
where the discrepancy converges to a minimum. Here,
the selected parameter value, P3, is selected with a β The connection models are developed as three-
valueof20%,thatis,atthepointwherethediscrepancy dimensional nonlinear FE models in ANSYS v.13.0.
is20%ofthetotalpossiblereductionindiscrepancy. As suggested by Kim et al. (2007), Bursi and Jaspart
Table 6 lists the EA term computed as a percentage (1997a,b, 1998),andSelametandGarlock(2010),both
of the nominal value, P0, according to Equations (2) themembersoftheconnectionsandtheboltsaremod-
and (3). For this application, the discrepancy tends to eled using three-dimensional solid finite elements (as
varyonalogarithmicscaleforparametervalues.There- opposed to simplified bolt models such as the coupled
fore, the percentages computed in Equation (3) are bolt model, spider bolt model, and no-bolt models as
modifiedtorepresentlog-basedpercentages.InTable6, discussedinKimetal.(2007)).SOLID187,a10-noded
theEAtermsareaveragedforeachconnection.Results tetrahedral element with quadratic displacement inter-
indicatethatthelargesterrorisassociatedwiththetop, polation is used to model steel members and bolts. Ef-
middle,andbaseconnections,respectively. fects of friction, represented using contact and target
elements available in ANSYS v.13.0, CONTA174, and
TARGET170, are modeled at all interfaces including:
5.3 CPIcalculation
beamtocolumn,bolttocolumn/beam,andboltholeto
The CPI values of the three connection models, calcu- bolt shank. The coefficient of friction is assumed for a
lated according to Equation (1), are listed in Table 7. ClassAsurfaceandsetto0.35(AISC,2008).Themesh
As the knowledge level for each connection stiffness size for all elements is selected through a mesh refine-
is low (i.e., constituents are highly uncertain), the UA ment study to assure that the numerical discretization
terms for each connection are set to one. The SA and is sufficiently fine to properly capture the deformation
EA terms are normalized such that the highest value and stress concentration in critical regions. Figure 11
in each respective category is one. This normalization presentstheFEmodelsforallthreeconnections.
bounds all three terms in the CPI calculation between As suggested by Citipitioglu et al. (2002), Kim et al.
zeroandone. (2007),andPirmozetal.(2008,2010),pretensioningof
Table 7 prioritizes the three constituents in the fol- theboltsisincludedinthemodel.However,asopposed
lowingorderofimportance:middle,base,andtopsug- to using temperature or initial displacement methods
gesting that the available resources should be devoted commonly found in the literature, pretension elements
toobtainanimprovedrepresentationofthemiddlecon- availableinANSYSv.13.0,PRETS179,thatdirectlyap-
nectionfirstandthebaseconnectionnext.Accordingly, ply a pretensioning force to the bolt volumes are im-
three FE models with gradually improved representa- plemented. This pretensioning force is determined by
tion of connections are developed in three successive converting the torque applied on the bolts using