Modal Pushover Analysis:Symmetric Symmetric- and Unsymmetric Unsymmetric-PlanBuildings
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Transcript of Modal Pushover Analysis:Symmetric Symmetric- and Unsymmetric Unsymmetric-PlanBuildings
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Bled.1Seismic Demands for Performance-Based Engineering
Modal Pushover Analysis: Modal Pushover Analysis: SymmetricSymmetric-- and Unsymmetricand Unsymmetric--Plan Plan
BuildingsBuildings
Anil K. ChopraRakesh K. Goel
International Workshop on Performance-Based Seismic Design
Bled, Slovenia28 June – 1 July 2004
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Bled.2Seismic Demands for Performance-Based Engineering
Improved Nonlinear Static ProcedureImproved Nonlinear Static Procedure
GoalsGoals
Retain the conceptual simplicity and computational attractiveness of current NSP
Obtain much improved estimate of seismic demands
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Bled.3Seismic Demands for Performance-Based Engineering
Response History AnalysisResponse History AnalysisSymmetricSymmetric--plan Buildingsplan Buildings
Equations of motion:
Solve directly these coupled equations
( ) ( ),signs gu t= −mu +cu +f u u m&& & & &&ι
Spatial (height-wise) distribution of forces
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Bled.4Seismic Demands for Performance-Based Engineering
Modal Expansion of Force DistributionModal Expansion of Force Distributionι=s m
nn n= = Γ∑ ∑ ms s φ
L T Tn L Mn n n n n nMnΓ = = =m mφ ι φ φ
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Bled.5Seismic Demands for Performance-Based Engineering
NineNine--Story SAC BuildingStory SAC Building
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Bled.6Seismic Demands for Performance-Based Engineering
Natural Vibration Periods and ModesNatural Vibration Periods and ModesNine-story SAC building
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Bled.7Seismic Demands for Performance-Based Engineering
Modal Expansion of Forces, sModal Expansion of Forces, sNine-story SAC building
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Bled.8Seismic Demands for Performance-Based Engineering
Response History AnalysisResponse History AnalysisUnsymmetricUnsymmetric--plan Buildingsplan Buildings
Equations of motion:
Solve directly these coupled equations
( ) ( ) ( ),sign gx gyu t u ts⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪− −⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭
=m1 00 m10 0
Mu +f u u && &&&& &
Spatial distribution of forces s
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Bled.9Seismic Demands for Performance-Based Engineering
Modal Expansion of Force Modal Expansion of Force Distribution Distribution s
O
xn
n n yn
nθ
⎧ ⎫⎪ ⎪
= = Γ ⎨ ⎬⎪ ⎪⎩ ⎭
∑ ∑m
s s m
I
φφ
φ
( )( )
for
for
ΤgxxnΤn
n n n n n Τn gyyn
u tL M LM u t
⎧⎪Γ = = = ⎨⎪⎩
m1M
m1
&&
&&
φφ φ
φ
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Bled.10Seismic Demands for Performance-Based Engineering
Modal Expansion of Forces, sModal Expansion of Forces, snn n= = Γ∑ ∑s s φΜ
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Bled.11Seismic Demands for Performance-Based Engineering
Modal Analysis ConceptsModal Analysis Concepts
“Modal” expansion of forces:
• Contribution of nth-”mode” to s and :
Response to ?
ι=s m
nn n= = Γ∑ ∑m msι φ
( ) ( ),effn n n n n gt u t= Γ = −s m p s &&φ
( )eff tp
( )eff,n tp
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Bled.12Seismic Demands for Performance-Based Engineering
Numerical Confirmation: Elastic SystemNumerical Confirmation: Elastic System
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Bled.13Seismic Demands for Performance-Based Engineering
Numerical Confirmation: Inelastic SystemNumerical Confirmation: Inelastic System
Other “modes” start responding after yielding begins
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Bled.14Seismic Demands for Performance-Based Engineering
Modal Analysis ConceptsModal Analysis Concepts“Modal” expansion of
• Contribution of nth-”mode” to s and :
Response to ?
s
( ) ( )eff ,
O
xn xnn yn n yn n n g
n n
t u t
θ θ
⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪
= = Γ = −⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭
s ms s m p s
s I
&&
φφ
φ
( )eff tp
( )eff,n tp
O
xn
n n yn
nθ
= = Γ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
∑ ∑m
s s m
I
φ
φ
φ
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Bled.15Seismic Demands for Performance-Based Engineering
Confirmation: Unsymmetric System, U1Confirmation: Unsymmetric System, U1
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Bled.16Seismic Demands for Performance-Based Engineering
Modal Pushover Analysis (MPA)Modal Pushover Analysis (MPA)
Estimate peak “modal” response of structure to
by
Pushover analysis for force distribution up to roof displacement
Combine peak “modal” responses(SRSS or CQC)
nr
( ) ( )eff, n gn t u t= −p s &&
rnu
* or
O
xn
rn n rn n n n yn
n
u D
θ
φ= Γ =
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
ms m m
φφ φ
φΙ
*ns
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Bled.17Seismic Demands for Performance-Based Engineering
Modal Pushover Analysis (MPA)Modal Pushover Analysis (MPA)
For Elastic BuildingsMPA is identical to RSA
For Inelastic BuildingsMPA is motivated by the weak modal coupling of response to ( )gu tn-s &&
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Bled.18Seismic Demands for Performance-Based Engineering
Plastic Rotations from Total Story Plastic Rotations from Total Story Drifts (Gupta & Krawinkler)Drifts (Gupta & Krawinkler)
Story plastic drift = total drift – yield drift
Relate beam plastic rotations to story plastic drift
Simplifying assumptions necessary to estimate story yield drift
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Bled.19Seismic Demands for Performance-Based Engineering
Nonlinear BeamNonlinear Beam--Column ElementColumn Element
Bending moments from total rotations
( )4 2I I JEIM MypL
α θ θ= + +
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Bled.20Seismic Demands for Performance-Based Engineering
MPA vMPA v’’s NLs NL--RHA: Boston BuildingsRHA: Boston Buildings
Story Drifts:
First “mode” alone is inadequate
Including more modes improves results
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Bled.21Seismic Demands for Performance-Based Engineering
MPA vMPA v’’s NLs NL--RHA: Seattle BuildingsRHA: Seattle Buildings
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Bled.22Seismic Demands for Performance-Based Engineering
MPA vMPA v’’s NLs NL--RHA: L.A. BuildingsRHA: L.A. Buildings
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Bled.23Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss NLNL--RHA: Plastic RotationsRHA: Plastic RotationsFirst “mode” alone is inadequateHigher “modes” produce hinges in upper stories
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Bled.24Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.25Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.26Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.27Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.28Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.29Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts
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Bled.30Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Hinge RotationsFEMA: Hinge Rotations
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Bled.31Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Hinge RotationsFEMA: Hinge Rotations
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Bled.32Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Hinge RotationsFEMA: Hinge Rotations
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Bled.33Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss FEMA: Hinge RotationsFEMA: Hinge Rotations
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Bled.34Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss NLNL--RHA: L.A. BuildingRHA: L.A. Building
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Bled.35Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss NLNL--RHA: L.A. BuildingRHA: L.A. Building
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Bled.36Seismic Demands for Performance-Based Engineering
MPA MPA vv’’ss NLNL--RHA: L.A. BuildingRHA: L.A. Building
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Bled.37Seismic Demands for Performance-Based Engineering
MPA Is Less Accurate for System U2MPA Is Less Accurate for System U2
Plausible Reasons
Close modal periods, strong coupling of lateral and torsional motions in each mode
Roof displacement due to selected ground motion is considerably underestimated in MPA
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Bled.38Seismic Demands for Performance-Based Engineering
MPA: CQC and ABSSUM RulesMPA: CQC and ABSSUM Rules
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Bled.39Seismic Demands for Performance-Based Engineering
ClosureClosure
Inelastic behavior is now explicitly considered
Current standard methods require major improvement
Several approaches in development worldwide
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Bled.40Seismic Demands for Performance-Based Engineering
ClosureClosure
This presentation has emphasized one possible approach that
Gives considerably improved estimates of seismic demands
Retains the conceptual simplicity and computational attractiveness of standard procedures