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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
11
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
3
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
21
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
22
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)
23
Moment-rotation relation of plastic hinges
M
θA
BC
D E
24
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)
5
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
26
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 =
27
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
28
Part 2: Modal Pushover Analysis (MPA)
29
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)
6
31
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
32
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+ + = −
33
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
34
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
35
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= =
= =∑ ∑
36
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
37
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
38
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.
39
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
40
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.
41
−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)
44
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
45
(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
46
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= =
= =∑ ∑
47
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=∑
48
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
49
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
50
Evaluation of Modal Pushover Analysis (MPA) Procedure
51
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
52
Generic one-bay frames
3-story 6-story 9-story 12-story 15-story 18-story
53
Beam-hinge model
54
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
55
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 =
56
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)