Modal Pushover Analysis: Symmetric- and Unsymmetric-Plan Buildings Anil K. Chopra Rakesh K. Goel International Workshop on Performance-Based Seismic Design Bled, Slovenia 28 June – 1 July 2004 Seismic Demands for Performance-Based Engineering
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Improved Nonlinear Static Procedure Goals z Retain the conceptual simplicity and computational attractiveness of current NSP z Obtain much improved estimate of seismic demands
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Response History Analysis Symmetric-plan Buildings z Equations of motion:
&& + cu& + f s ( u,sign u& ) = 竏知ﾎｹ u&&g ( t ) mu Spatial (height-wise) distribution of forces
z Solve directly these coupled equations
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Modal Expansion of Force Distribution
s = mι s = ∑ sn = ∑ Γnmφ n Ln Γn = Mn
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Ln = φnT mι
M n = φnT mφ n
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Nine-Story SAC Building
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Natural Vibration Periods and Modes z Nine-story SAC building
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Modal Expansion of Forces, s z Nine-story SAC building
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Response History Analysis Unsymmetric-plan Buildings z Equations of motion:
&& + f s ( u,sign u& ) Mu
⎧m1⎫ ⎧ 0 ⎫ = − ⎪⎨ 0 ⎪⎬ u&&gx ( t ) − ⎪⎨m1⎪⎬ u&&gy ( t ) ⎪ 0 ⎪ ⎪ 0 ⎪ ⎩ ⎭ ⎩ ⎭
Spatial distribution of forces s
z Solve directly these coupled equations Seismic Demands for Performance-Based Engineering
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Modal Expansion of Force Distribution s ⎧ mφ xn ⎫ ⎪ ⎪ s = ∑ s n = ∑ Γ n ⎨ mφ yn ⎬ ⎪ ⎪ ⎩I Oφ θ n ⎭ Ln Γn = Mn
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M n = φnΤ Mφn
Τ ⎧φxn ⎪ m1 for u&&gx ( t ) Ln = ⎨ Τ φ ⎪⎩ ynm1 for u&&gy ( t )
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Modal Expansion of Forces, s s = ∑ sn = ∑ Γ nΜ φ n
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Modal Analysis Concepts z “Modal” expansion of forces: s = mι mι =
•
∑ s n = ∑ Γ nm φ n
Contribution of nth-”mode” to s and peff ( t ):
s n = Γ n mφ n
peff ,n ( t ) = −s n u&&g ( t )
z Response to peff,n ( t ) ?
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Numerical Confirmation: Elastic System
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Numerical Confirmation: Inelastic System
Other “modes� start responding after yielding begins Seismic Demands for Performance-Based Engineering
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Modal Analysis Concepts ⎧ mφ xn ⎫ ⎪ ⎪ s = ∑ s n = ∑ Γ n ⎨ mφ yn ⎬ ⎪ ⎪ ⎩I Oφ θ n ⎭
z “Modal” expansion of s
•
Contribution of nth-”mode” to s and peff ( t ) :
⎧s xn ⎫ ⎧ mφ xn ⎫ ⎪ ⎪ ⎪ ⎪ s n = ⎨s yn ⎬ = Γ n ⎨ mφ yn ⎬ ⎪ ⎪ ⎪ ⎪ φ s I ⎩ θn⎭ ⎩ O θn⎭
peff ,n ( t ) = −s n u&&g ( t )
z Response to peff,n ( t ) ? Seismic Demands for Performance-Based Engineering
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Confirmation: Unsymmetric System, U1
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Modal Pushover Analysis (MPA) z Estimate peak “modal” response rn of structure to by
peff,n ( t ) = −s nu&&g ( t )
z Pushover analysis for force distribution s*n up to roof displacement urn
urn = Γ nφrn Dn
s*n = m φ n
or
z Combine peak “modal” responses (SRSS or CQC) Seismic Demands for Performance-Based Engineering
⎧ mφ xn ⎫ ⎪ ⎪ ⎨mφ yn ⎬ ⎪ ⎪ ⎩ Ι Oφ θ n ⎭
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Modal Pushover Analysis (MPA) z For Elastic Buildings ĂŽMPA is identical to RSA
z For Inelastic Buildings ĂŽMPA is motivated by the weak modal coupling of response to -snu&&g ( t )
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Plastic Rotations from Total Story Drifts (Gupta & Krawinkler) z Story plastic drift = total drift – yield drift z Relate beam plastic rotations to story plastic drift z Simplifying assumptions necessary to estimate story yield drift
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Nonlinear Beam-Column Element z Bending moments from total rotations M I = Myp + α
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EI ( 4θ I + 2θ J ) L
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MPA v’s NL-RHA: Boston Buildings Story Drifts: zFirst “mode” alone is inadequate zIncluding more modes improves results Seismic Demands for Performance-Based Engineering
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MPA v’s NL-RHA: Seattle Buildings
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MPA v’s NL-RHA: L.A. Buildings
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MPA v’s NL-RHA: Plastic Rotations z First “mode” alone is inadequate z Higher “modes” produce hinges in upper stories
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Story Drifts
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MPA v’s FEMA: Hinge Rotations
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MPA v’s FEMA: Hinge Rotations
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MPA v’s FEMA: Hinge Rotations
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MPA v’s FEMA: Hinge Rotations
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MPA v’s NL-RHA: L.A. Building
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MPA v’s NL-RHA: L.A. Building
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MPA v’s NL-RHA: L.A. Building
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MPA Is Less Accurate for System U2 zPlausible Reasons ĂŽ Close modal periods, strong coupling of lateral and torsional motions in each mode ĂŽ Roof displacement due to selected ground motion is considerably underestimated in MPA Seismic Demands for Performance-Based Engineering
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MPA: CQC and ABSSUM Rules
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Closure zInelastic behavior is now explicitly considered zCurrent standard methods require major improvement zSeveral approaches in development worldwide
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Closure zThis presentation has emphasized one possible approach that ĂŽGives considerably improved estimates of seismic demands ĂŽRetains the conceptual simplicity and computational attractiveness of standard procedures
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