The document summarizes research using multi-modal response spectrum analysis implemented in the OpenSees software to analyze the seismic behavior of existing bridges. It describes using response spectrum analysis to conduct a parametric study of thousands of bridges to identify vulnerable configurations. Examples are presented of analyzing bridges with nonlinear isolator elements, assessing an existing bridge, and parametrically varying properties of multi-girder bridges to determine critical components. Response spectrum analysis is shown to provide an upper limit on responses and enable rapid analysis suitable for parametric studies and seismic assessments.

1. Budapest University of Technology and Economics
Department of Structural Engineering
Multi modal response spectrum analysis implemented in OpenSEES
OpenSeesDays Portugal
July03-04, 2014, Porto
József Simon
László Gergely Vigh

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Introduction
Research project:
Seismic analysis of bridges in moderate seismic regions
Database of existing bridges
Pre-Eurocode era –lack of seismic design
Seismic behavior is not known
Thousands of structures
Parametric study to recognize vulnerable configurations
Fast analysis method is needed
Multi modal response spectrum analysis

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Introduction
Response spectrum analysis:
Mostwidelyused
Well-known
Relativelyfast(comparedtoothermethods)
Suitableforparametricstudyorsensitivityanalysis
Linearmethod,but
Canbeusedwithnon-linearelements(effectivestiffnessanditerativesolution)
Overestimates(generally)theinternalforces
Thus,givesanupperlimitfortheresponses

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풑푖 = 풎 휱푖
휱푖
푇풎 휾
휱푖
푇풎 휱푖
푆푑 푇푖
Theoretical background
Response spectrum analysis:
• Based on modal analysis (eigenvectors and eigenvalues)
• Importance of a mode characterized by the modal mass:
푚푖
∗ =
휱푖
푇풎 휾 2
휱푖
푇풎 휱푖
• Load vector for each mode:
• Quasi-static loading, static analysis
• Combination of the responses from each mode:
퐸퐴퐵푆푆푈푀 = 퐸푖 , 퐸푆푅푆푆 = 퐸푖 ퟐ , 퐸퐶푄퐶 = 퐸푖 휌푖푗 퐸푗
풊 풋

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RSA modul in OpenSees
proc RSA{agsoiltype eqtype ξ q qdsumtype elelist nodelist}
WritteninTCL
3Danalysisofanystructure
agpeakgroundacceleration[m/s2]
soiltype(A,B,C,DorE)
eqtype(1or2)
ξdampingratio
qbehaviorfactor
qdneededtocalculatedisplacements
sumtype(ABSSUM,SRSS,CQC)comb.method
elelist
nodelist
Defines the shape of the standard spectrum
Only results of listed nodes and elements are stored and combined

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RSA modul in OpenSees
Modalanalysis>eigencommand(Φi)
Massmatrixcompilation>nodeMasscommand(m)
Note: mass has to be defined in nodes!
Modalmasscalculation(m* i)
Necessarynumberofmodes(e.g.90%rule)
Loadvectorcompilation(pi)
Staticanalyses
Storingresponses(nodelistandeleslist)
Note: only eleResponse$e_num forces!
Combinationofresponses(sumtype)
Combinedresponses>globalvariables

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RSA modul in OpenSees
Output:
Globalvariables
Globalcoordinatesystem!!(displacementsandforcesaswell)
3·3=9globallistvariablesfordisplacements
3·2·6=36globallistvariablesforelementinternalforces

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Ex.1 -Equivalent linear analysis of girder bridges with seismic isolators
EC8-2Section7
1.Non-linearbehaviorapproximatedbybi-linearcharacteristic.
NLTHA
2.Equivalentlinearstiffnessandeffectivedamping.
LRSA
ITERATION NEEDED!!
ξ = ??
•EC -AASHTO
•JPWRI
•CALTRANS 94
•CALTRANS 96
Fivestandard methods:
Continousgirderbridges
Withisolationbearinginthelongitudinaldirection
Comparisonoftwoanalysismethods(non-linearelements!)

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Ex.1 -Equivalent linear analysis of girder bridges with seismic isolators
ThreedimensionalnumericalmodelinOpenSees
No energy dissipation is expected in the piers!!

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Ex.1 -Equivalent linear analysis of girder bridges with seismic isolators
In TCL (automatic iteration) with RSA

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Ex.1 -Equivalent linear analysis of girder bridges with seismic isolators
Threecontinousgirderbridges(concrete,composite,steel)
Onlyonefixed/isolationbearing!
Twosoiltypes(C,E),twopeakgroundacc.(1.0and1.4m/s2)
C
ag=1.4 m/s2
Compared to NLTHA:
The accuracy of the internal forces (pier moments, isolator forces) is ±25%

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Ex.2 - Seismic assessment of M0 Highway bridge at Háros, Hungary
0
50
100
150
200
250
300
350
P2 P3 P4 P5 P6 P7 P8 P9
Mx longitudinal moments [MNm]
Pier number
RSA
THA
Comparison of RSA and LTHA
results
Upper-limit for the responses!

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Ex.3 –Parametric seismic analysis of multi-girder bridges
Number of supports:2-5
Spans: 4-8-12-16-20-24 m
Outer/inner span ratio: 0.75-1.00
Pier height:2-6-10-14 m
Pier cap cross-section:1.20x1.00 m
Pier:0.60x0.90 m
Abutment:1.00x2.00 m (14 m wide)
Number of piers in the lateral direction:4 (distributed in every 3 m)
Piles:D=80 cm 1x5 (abutment) –2x5 (piers)
Fixed parameters:
Nearly half of the higher level road bridges!!!

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Superstructure: ElasticBeamColoumn
Pier: ElasticBeamColoumn
Abutments: ElasticBeamColoumn
Rigid links
Rigid links
Foundation and backfill: linear springs
Ex.3 –Parametric seismic analysis of multi-girder bridges

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Ex.3 – Parametric seismic analysis of multi-girder bridges
Superstructure
Abutments
Backfill soil
Superstructure-abutment
Superstructure-pier joints
Piers
Pier foundations
Hp (m) 4 8 12 16 20 24
2 0.56 1.13 1.77 2.11 3.04 3.73
6 0.47 0.85 1.27 1.75 2.27 2.98
10 0.64 0.94 1.26 1.63 2.10 2.68
14 1.31 1.60 1.80 2.15 2.62 3.05
Támaszköz méterben
Critical components at longer
bridges
Components found to be adequate
Spans (m)
D/C ratio for piers
(4 span bridges)

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Concluding remarks
Response spectrum analysis:
Linearmethod,but
Canbeusedwithnon-linearelements(Ex.1)
Overestimates(generally)theinternalforces
Thus,givesanupperlimitfortheresponses(Ex.2)
Relativelyquick,
Suitableforparametricstudyorsensitivityanalysis(Ex.3)
Alsoitprovidessolidbasefor:
detailedseismicassessment
retrofitplanning
developmentofdesignconceptsfornewbridges

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Concluding remarks
RSA should be implemented in the soure code:
Massmatrixcanbeconstructedevenwithdiscontinuous, unlumpedmasses
Duringthecombinationoftheresponses,theelementtypecanberecognizedandspecificresponsescanbequeried, stored
Anystructurecanbeanalyzedin2Dor3Daswell

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Thank you for your attention!
Acknowledgement :
This presentationwas supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.