1

1

Pushover Analysis for Seismic Evaluation of buildings

Dr. Chatpan ChintanapakdeeJuly 3, 2003

2

Outline

Part 1: Conventional pushover analysis• Static pushover analysis• Seismic evaluation• FEMA nonlinear static procedure (NSP)

Part 2: Modal pushover analysis (MPA)• Dynamic analysis procedures• Evaluation of MPA

3

Part 1: Conventional Pushover Analysis

4

Static pushover analysis

is static inelastic analysis of a structure subjected to monotonically increasing lateral forces with an invariant height-wise distribution, i.e., increasing load factor while fixing load pattern

tells sequence and magnitudes of yielding (damage), internal forces, deformations, and failure mechanism

5 6

Static pushover analysis

Pushover analysis is similar to the plastic analysis, where failure mechanism and collapse load factor is determined and the moment-rotation relation of plastic hinge is only rigid-plastic.

But, pushover analysis also keeps track of structural response as the load factor increases incrementally and moment-rotation relation of plastic hinge can be more complicated.

2

7

Structural model for pushover analysis

Structural model consists of nonlinear elements

Nonlinear element can be for any type of forces, e.g., bending, shear, or axial force

Plastic hinge is a simple nonlinear element to model yielding in bending

Information on moment-rotation or moment-curvature relation of plastic hinge is required

SAP2000 is a program that has plastic hinge element

8

Plastic-hinge model

9

Force-deformation relation (or Moment-rotation relation )

Force-deformation relation is no longer linear elasticKnowledge of cyclic behavior is not necessary in pushover analysis; only the first loading branch is requiredForce-deformation relation can be elastoplastic, bilinear, degrading, etcMoment-rotation relation is often rigid-plastic

10

Force-deformation relation

Elastoplastic Bilinear Degrading

Rigid-plastic with/without post-yield stiffness

M M

θ θ

F F F

D D D

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Height-wise Force Distributions(Force Pattern)

Uniform 1st Mode Shape

12

Pushover curve

is a plot of base shear versus roof displacementshows nonlinear behavior of the buildingis usually idealized by bilinear curve

Note that global yield point not the same as first local yield point

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13

Base shear

Roof displacement

Pushover curve

Bilinear idealization

Actual curve

14

Seismic Evaluation using Pushover Analysis

Seismic evaluation is to assess the seismic performance of a structure by comparing the seismic demands to capacities

Pushover analysis is used to approximatelydetermine the structural responses (seismic demand) due to an earthquake ground motion (or average response due to a set of earthquake ground motions)

15

Seismic DemandsInternal forces

Displacement ui of the ith story relative to the ground

Inter-story drift ∆i of the ith story = ui - ui-1

u1

u2

2∆

ug 16

Pushover analysis to determine seismic demands

The seismic demands are computed by nonlinear static analysis of the structure subjected to monotonically increasing lateral forces with an invariant height-wise distribution until a predetermined target displacement is reached.

The target roof displacement is determined from the deformation of an equivalent single-degree-of-freedom (SDF) system due to the earthquake ground motion

17

Assumptions

1. The response of the multi-degree-of-freedom (MDF) structure can be related to the response of an equivalent SDF system, implying that the response is controlled by a single mode and this mode shape remains unchanged even after yielding occurs

2. The invariant lateral force distribution can represent and bound the distribution of inertia forces during an earthquake

18

Equivalent inelastic single-degree-of-freedom (SDF) system

Force-displacement relation of SDF system is determined from pushover curve (base shear-roof displacement)

4

19

(a) Idealized Pushover Curve

ur n

Vbn

ur n y

Vbny

Actual

Idealized

1k

n

1α

nk

n

(b) Fsn

/ Ln − D

n Relationship

Dn

Fsn

/ Ln

Dny

= ur n y

/ Γn φ

r n

Vbny

/ M*n

1ω

n2

1α

nω

n2

Explain more laterSee Chopra and Goel 2002

20

Response of SDF system to earthquake

Equation of motion is

( )2 ( , )s gD D F D D u tζω+ + = −

( )gu t = earthquake ground acceleration

( , )sF D D is determined from pushover curve

D = displacement of equivalent SDF system

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Target roof displacement

Target roof displacement is determined from displacement of equivalent SDF system

where Do = peak value of D

Seismic demands equal to response of structure from pushover analysis when roof displacement equal to target roof displacement

1 1ro r ou Dφ= Γ

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FEMA-273Nonlinear Static Procedure (NSP)

describes in detail how to use do seismic evaluation using pushover analysis

specifies moment-rotation relationship, force patterns, how to determine target roof displacement (coefficient method), acceptance criteria, and limitation of the procedure (when NSP should not be used)

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Moment-rotation relation of plastic hinges

M

θA

BC

D E

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FEMA-273 force distributions(force patterns)

RequiredUniform (acceleration)

And choose one or more of the followings:Equivalent lateral force (ELF) SRSS pattern = Lateral force back calculated from story shear determined by response spectrum analysis (RSA)First mode pattern (new in FEMA-356)

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25

FEMA force distributions

0.024

0.048

0.070

0.093

0.114

0.134

0.154

0.173

0.191

(a) 1st Mode

0.006

0.020

0.040

0.065

0.096

0.130

0.170

0.213

0.260

(b) ELF

0.049

0.086

0.096

0.079

0.050

0.044

0.087

0.183

0.326

(c) SRSS

0.111

0.111

0.111

0.111

0.111

0.111

0.111

0.111

0.111

(d) Uniform

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Target roof displacement

δt = Target roof displacement

C0 = Factor to relate spectral displacement to roof disp.

C1 = Factor to relate inelastic to elastic displacement

C2 = Factor to include degradation of hysteresis loop

C3 = Factor to include P-Delta effect

Sa = Elastic spectral acceleration

Te = Effective period

gTSCCCC eat 2

2

3210 4πδ =

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Limitation of NSP (FEMA-273: 2.9.2.1)

The NSP should not be used for structures in which higher mode effects are significant because it assumes that the response is controlled by a single (fundamental) mode

This leads to the development of modal pushover analysis (MPA), which includes the contribution of higher modes

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Part 2: Modal Pushover Analysis (MPA)

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Outline of Part 2Dynamic Analysis Procedures

Elastic buildingInelastic building

Evaluation of Modal Pushover Analysis (MPA) using generic frames

Structural systems and ground motionsNonlinear Response History Analysis resultsComparison of MPA and NL-RHA results

30

Dynamic analysis proceduresElastic building

Modal Response History Analysis (RHA)Response Spectrum Analysis (RSA)

Inelastic buildingNonlinear Response History Analysis (NL-RHA)Uncoupled Modal RHA (UMRHA)Modal Pushover Analysis (MPA)

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Elastic systems

Equations of Motions

m, c, k = mass, damping, and stiffness matrices

= ground acceleration, ι = [1 1 … 1]T

( )gu t+ + = −mu cu ku mι

( )gu t

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Modal Response History Analysis (RHA)

Equation of motions can be uncoupled using eigenvectors of mass and stiffness matrices, which are mode shapes φn

1 1

N N

n n n nn n

D= =

= = Γ∑ ∑u u φ

T Tnn n n n n n

n

L L MM

Γ = = =φ m φ mφι

( )22n n n n n n gD D D u tζ ω ω+ + = −

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Modal RHA

( ) ( )eff gt u t= −p mι1 1

N N

n n nn n= =

= = Γ∑ ∑m s mι φ

( ) ( ) ( )eff eff ,1 1

N N

n n gn n

t t u t= =

= = −∑ ∑p p s

−m 0 m0

1

2

3

4

5

6

Sto

ry

s

=

−m 0 m

+

s1

−m 0 m

+

s2

−m 0 m

+ ...

s3

• Effective earthquake force peff(t) can also be decomposed into modes

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Modal RHA

Key concepts

Response of each mode is controlled by the response Dn(t) of modal SDF system to üg(t)

Response un(t) is due to peff,n(t) and is proportional to the nth mode shape

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Modal RHA

For a response quantity, r, the total response

where rnst is the modal static response [Chopra,

2001]

2

1 1( ) ( ) ( )

N Nst

n n n nn n

r t r t r D tω= =

= =∑ ∑

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Response Spectrum Analysis (RSA)

The peak value of r(t) is estimated from

Square Root of Sum Square (SRSS)

where and

which can be obtained from deformation spectrum

2

1

N

o non

r r=∑

max ( )no ntD D t

∀=2st

no n n nor r Dω=

7

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Inelastic systems

Equation of motions

is the relations between lateral forces and displacements at floors, which is no longer equal to ku after yielding occurs

( , ) ( )s gu t+ + = −mu cu f u u mι

( , )sf u u

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Nonlinear Response History Analysis (NL-RHA)

This set of equations can be solved exactly only by numerical methods, e.g., Newmarkmethod [Chopra, 2001], widely known as nonlinear dynamic analysis or nonlinear time-history analysis.

The results of this analysis will be referred as the “exact” values.

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Uncoupled Modal RHA (UMRHA)

Trying to uncoupled this set of equations as for elastic system leads to

which remains coupled becausedepends on all D = [ D1 D2 …DN]T

( )( , )2 snn n n n g

n

FD D u tL

ζ ω+ + = −D D

( , )snF D D

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UMRHAIf we apply peff,n(t) excitation to an elasticsystem, use RHA to solve for the u(t) and decompose u(t) using elastic mode shapes, we find that all modes other than the nth modedo not participate.

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−5

0

5

−4.63

Decomposed Mode 1

u r1 (

cm)

peff,1

(t) = −s1ü

g(t)

(a) Elastic

−5

0

5

0.00

Decomposed Mode 2

u r2 (

cm)

0 10 20 30−5

0

5

0.00

Decomposed Mode 3

u r3 (

cm)

Time (sec)

−1

0

1

0.00

Decomposed Mode 1

u r1 (

cm)

peff,2

(t) = −s2ü

g(t)

−1

0

1

−0.63

Decomposed Mode 2

u r2 (

cm)

0 10 20 30−1

0

1

0.00

Decomposed Mode 3

u r3 (

cm)

Time (sec)

42

UMRHAIf we apply peff,n(t) excitation to an inelastic system, use nonlinear RHA to solve for the u(t) and decompose u(t) using elastic mode shapes, we find that all modes other than the nth mode do not participate significantly.

8

43

−5

0

5 3.14 Decomposed Mode 1

u r1 (

cm)

peff,1

(t) = −s1ü

g(t)

(b) Inelastic

−5

0

5

−0.10

Decomposed Mode 2

u r2 (

cm)

0 10 20 30−5

0

5

0.01

Decomposed Mode 3

u r3 (

cm)

Time (sec)

−1

0

1

0.28Decomposed Mode 1

u r1 (

cm)

peff,2

(t) = −s2ü

g(t)

−1

0

1 0.68 Decomposed Mode 2

u r2 (

cm)

0 10 20 30−1

0

1

0.09

Decomposed Mode 3

u r3 (

cm)

Time (sec)

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UMRHA

Making an approximation that only depends on Dn results in a set of uncoupled equations, which are equations of motions of inelastic SDF systems

where is determined from

pushover curve using mφn as force pattern

( , )snF D D

( )( , )2 sn n nn n n n g

n

F D DD D u tL

ζ ω+ + = −

( , )sn n n

n

F D DL

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(a) Idealized Pushover Curve

ur n

Vbn

ur n y

Vbny

Actual

Idealized

1k

n

1α

nk

n

(b) Fsn

/ Ln − D

n Relationship

Dn

Fsn

/ Ln

Dny

= ur n y

/ Γn φ

r n

Vbny

/ M*n

1ω

n2

1α

nω

n2

2*

*sny bny rny n

ny nn n n rn n

F V u LD ML M Mφ

= = =Γ

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UMRHA

The total response can be calculated as

2

1 1( ) ( ) ( )

N Nst

n n n nn n

r t r t r D tω= =

= =∑ ∑

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Modal Pushover Analysis (MPA)The modal peak response rno is obtained from pushover analysis corresponding to target roof displacement

The total response is obtained using SRSS

rno n nr nou Dφ= Γ

2

1

N

o non

r r=∑

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Step-by-Step Procedure of MPA

1. Determine mode shapes φn and frequencies

2. Perform pushover analysis using mφn as force pattern

3. Idealize pushover curve and determine properties of equivalent inelastic SDF system (Fsny/Ln and Dny)

4. Calculate D(t), Dno, and target roof displacement

rno n nr nou Dφ= Γ

9

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Step-by-Step Procedure of MPA

5. Obtain rno from pushover analysis corresponding to target roof displacement

6. Repeat step 2-5 for all “modes” to be included

7. Combine rno using SRSS rule to obtain total peak response ro

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Evaluation of Modal Pushover Analysis (MPA) Procedure

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Structural SystemsGeneric one-bay frames of six different height: 3, 6, 9, 12, 15, 18 stories

Ibeam=Icolumn tuned such that story drifts due to IBC force pattern [2000 International Building Code] are equal in all stories and T1=0.045H0.8 (H=height in ft.)

Plastic hinges form only at beam ends and base of 1st

story columns and yielding occurs simultaneously under IBC force pattern

All bending strength is scaled such that yield base shear is Vby=Ay*W/g where Ay is median constant-ductility inelastic spectrum for µ=1, 1.5, 2, 4, 6

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Generic one-bay frames

3-story 6-story 9-story 12-story 15-story 18-story

53

Beam-hinge model

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LMSR set of 20 ground motions

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

Pse

udo−

acce

lera

tion,

A/g

Natural vibration period, Tn (sec)

10

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Statistics of results

1ln

ˆ exp

n

ii

xx

n=

=

∑

( )1/ 2

2

1

ˆln ln

1

n

ii

x x

nδ =

−

=−

∑

Median value =

Dispersion measure =

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Nonlinear RHA results

0 0.5 1G

1

2

3

Sto

ry

(a) 3−story frames

Elasticµ= 1 1.5 2 4 6

0 0.5 1G

1

2

3

4

5

6(b) 6−story frames

0 0.5 1G123456789

(c) 9−story frames

0 0.5 1G

2

4

6

8

10

12

Sto

ry

Story drift (%)

(d) 12−story frames

0 0.5 1G

3

6

9

12

15

Story drift (%)

(e) 15−story frames

0 0.5 1G2468

1012141618

Story drift (%)

(f) 18−story frames

57

Comparison of MPA and NL-RHA

0 0.2 0.4 0.6 0.8G

1

2

3

Sto

ry

Elastic

3−storyframe

0 0.2 0.4 0.6 0.8G

1

2

3µ=2

0 0.2 0.4 0.6 0.8G

1

2

3µ=4

0 0.2 0.4 0.6 0.8G

1

2

3µ=6

NL−RHA MPA 1 "mode" 2 "modes"3 "modes"

0 0.2 0.4 0.6 0.8G

1

2

3

4

5

6

Sto

ry

6−storyframe

0 0.2 0.4 0.6 0.8G

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8G

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8G

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8G123456789

Sto

ry

Story drift (%)

9−storyframe

0 0.2 0.4 0.6 0.8G123456789

Story drift (%)0 0.2 0.4 0.6 0.8

G123456789

Story drift (%)0 0.2 0.4 0.6 0.8

G123456789

Story drift (%)

58

Comparison of MPA and NL-RHA

0 0.2 0.4 0.6 0.8G

2

4

6

8

10

12

Sto

ryElastic

12−storyframe

0 0.2 0.4 0.6 0.8G

2

4

6

8

10

12µ=2

0 0.2 0.4 0.6 0.8G

2

4

6

8

10

12µ=4

0 0.2 0.4 0.6 0.8G

2

4

6

8

10

12µ=6

NL−RHA MPA 1 "mode" 2 "modes"3 "modes"

0 0.2 0.4 0.6 0.8G

3

6

9

12

15

Sto

ry

15−storyframe

0 0.2 0.4 0.6 0.8G

3

6

9

12

15

0 0.2 0.4 0.6 0.8G

3

6

9

12

15

0 0.2 0.4 0.6 0.8G

3

6

9

12

15

0 0.2 0.4 0.6 0.8G2468

1012141618

Sto

ry

Story drift (%)

18−storyframe

0 0.2 0.4 0.6 0.8G2468

1012141618

Story drift (%)0 0.2 0.4 0.6 0.8

G2468

1012141618

Story drift (%)0 0.2 0.4 0.6 0.8

G2468

1012141618

Story drift (%)

59

Bias of MPA

0.5 1 1.5G

1

2

3

Sto

ry

(a) 3−story frames

Elasticµ= 1 1.5 2 4 6

0.5 1 1.5G

1

2

3

4

5

6(b) 6−story frames

0.5 1 1.5G123456789

(c) 9−story frames

0.5 1 1.5G

2

4

6

8

10

12

Sto

ry

Story−drift ratio ∆*MPA

or ∆*RSA

(d) 12−story frames

0.5 1 1.5G

3

6

9

12

15

Story−drift ratio ∆*MPA

or ∆*RSA

(e) 15−story frames

0.5 1 1.5G2468

1012141618

Story−drift ratio ∆*MPA

or ∆*RSA

(f) 18−story frames

60

MPA versus NL-RHA

0 1 20

1

2(a) µ=1, 12th story

∆ MP

A (%

)

Story 12∆*

MPA=0.765

δ=0.145

0 1 20

1

2(b) µ=2, 12th story

∆*MPA

=0.735

δ=0.213

0 1 20

1

2(c) µ=4, 12th story

∆*MPA

=1.034

δ=0.358

0 1 20

1

2(d) µ=6, 12th story

∆*MPA

=1.282

δ=0.383

0 1 20

1

2(e) µ=1, 6th story

∆ MP

A (%

)

Story 6∆*

MPA=0.982

δ=0.131

0 1 20

1

2(f) µ=2, 6th story

∆*MPA

=1.035

δ=0.252

0 1 20

1

2(g) µ=4, 6th story

∆*MPA

=0.966

δ=0.249

0 1 20

1

2(h) µ=6, 6th story

∆*MPA

=1.000

δ=0.166

0 1 20

1

2(i) µ=1, 1st story

∆ MP

A (%

)

Story 1

∆NL−RHA

(%)

∆*MPA

=1.085

δ=0.179

0 1 20

1

2(j) µ=2, 1st story

∆NL−RHA

(%)

∆*MPA

=1.126

δ=0.159

0 1 20

1

2(k) µ=4, 1st story

∆NL−RHA

(%)

∆*MPA

=0.891

δ=0.137

0 1 20

1

2(l) µ=6, 1st story

∆NL−RHA

(%)

∆*MPA

=0.838

δ=0.138

11

61

Dispersion of MPA estimates

0 0.1 0.2 0.3 0.4G

1

2

3(a) 3−story frames

Sto

ry Elasticµ= 1 1.5 2 4 6

0 0.1 0.2 0.3 0.4G

1

2

3

4

5

6(b) 6−story frames

0 0.1 0.2 0.3 0.4G123456789

(c) 9−story frames

0 0.1 0.2 0.3 0.4G

2

4

6

8

10

12(d) 12−story frames

Sto

ry

Dispersion of ∆*MPA

or ∆*RSA

0 0.1 0.2 0.3 0.4G

3

6

9

12

15(e) 15−story frames

Dispersion of ∆*MPA

or ∆*RSA

0 0.1 0.2 0.3 0.4G2468

1012141618

(f) 18−story frames

Dispersion of ∆*MPA

or ∆*RSA

62

Conclusions

Including the response contribution due to the second “mode” leads to story drifts that are much more accurate than based on first “mode”alone, which is the basis for procedure currently used in structural engineering practice. The third “mode” contribution should also be included for taller frames.

63

Conclusions

While errors in the MPA procedure tend to increase for longer-period frames and larger SDF-system ductility factors, these trends are not perfect.

64

Conclusions

The RSA procedure consistently underestimates the response of elastic structures. The bias and dispersion in MPA estimates of seismic demands for inelastic frames are usually larger than for elastic systems (determined by RSA procedure).

65

The End

Questions?

66

ReferencesKrawinkler and Sereviratna 1998 (Engineering and Structures Journal)Chopra and Goel 2001 (PEER Report)Chopra and Goel 2002 (EESD Journal)Chintanapakdee and Chopra 2003 (EESD Journal)Chintanapakdee and Chopra 2003 (UCB/EERC Report)