modal pushover analysis:symmetric symmetric- and unsymmetric unsymmetric-planbuildings

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


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


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


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


NineNine--Story SAC BuildingStory SAC Building


Natural Vibration Periods and ModesNatural Vibration Periods and ModesNine-story SAC building


Modal Expansion of Forces, sModal Expansion of Forces, sNine-story SAC building


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


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


Modal Expansion of Forces, sModal Expansion of Forces, snn n= = Γ∑ ∑s s φΜ


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


Numerical Confirmation: Elastic SystemNumerical Confirmation: Elastic System


Numerical Confirmation: Inelastic SystemNumerical Confirmation: Inelastic System

Other “modes” start responding after yielding begins


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


Confirmation: Unsymmetric System, U1Confirmation: Unsymmetric System, U1


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


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 &&


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


Nonlinear BeamNonlinear Beam--Column ElementColumn Element

Bending moments from total rotations

( )4 2I I JEIM MypL

α θ θ= + +


MPA vMPA v’’s NLs NL--RHA: Boston BuildingsRHA: Boston Buildings

Story Drifts:

First “mode” alone is inadequate

Including more modes improves results


MPA vMPA v’’s NLs NL--RHA: Seattle BuildingsRHA: Seattle Buildings


MPA vMPA v’’s NLs NL--RHA: L.A. BuildingsRHA: L.A. Buildings


MPA MPA vv’’ss NLNL--RHA: Plastic RotationsRHA: Plastic RotationsFirst “mode” alone is inadequateHigher “modes” produce hinges in upper stories


MPA MPA vv’’ss FEMA: Story DriftsFEMA: Story Drifts


MPA MPA vv’’ss FEMA: Hinge RotationsFEMA: Hinge Rotations


MPA MPA vv’’ss NLNL--RHA: L.A. BuildingRHA: L.A. Building


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


MPA: CQC and ABSSUM RulesMPA: CQC and ABSSUM Rules


ClosureClosure

Inelastic behavior is now explicitly considered

Current standard methods require major improvement

Several approaches in development worldwide


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

modal pushover analysis:symmetric symmetric- and unsymmetric unsymmetric-planbuildings

Documents

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effn n n n n gt u t

s mnn n

n tpoxnn n ynn

n tpbled

s ms s

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