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Farzad Naeim Structural Dynamics for Practicing Engineers 1of 71
(Last Revision Date: 5-18-2009)
Part II:Analysis of Nonlinear
Structural Response
Farzad Naeim, Ph.D., S.E., Esq.Farzad Naeim, Ph.D., S.E., Esq.Vice President and General CounselVice President and General Counsel
J ohn A. Martin & Associates, Inc.J ohn A. Martin & Associates, Inc.
Farzad Naeim Structural Dynamics for Practicing Engineers 2of 71
(Last Revision Date: 5-18-2009)
Single-degree-of-freedom systemsubjected to time-dependent force.
Static Equilibrium:
Dynamic Equilibrium:
Three Simplifying Assumptions
for SDOF:1. Mass concentrated at the roof2. Roof is Rigid3. Axial Deformation of Columns
Neglected
kvp
)()()()( tvtktvctvmtp
STATIC AND DYNAMIC EQUILIBRIUM
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Farzad Naeim Structural Dynamics for Practicing Engineers 3of 71
(Last Revision Date: 5-18-2009)
Single-degree-of-freedom systemsubjected to base motion.
)(tgmtvtktvtctvm
)(tgm
Response of a SDOFsystem to earthquakeground motion:
STATIC AND DYNAMIC EQUILIBRIUM
In reality, if parts of thestructure collapses, orthere is pounding withanother structure, evenmass could be a
function of time.
Farzad Naeim Structural Dynamics for Practicing Engineers 4of 71
(Last Revision Date: 5-18-2009)
TWO TYPES OF NONLINEARITY
MaterialMaterial
GeometricGeometric
(caused by large deformations)(caused by large deformations)
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Farzad Naeim Structural Dynamics for Practicing Engineers 5of 71
(Last Revision Date: 5-18-2009)
ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS (MDOF)
Analogous to the case of SDOF systems:
)(}]{[}]{[}]{[ tgMvKvCvM Influence vector
In general case:
n
n
nn
nnnn
nn
nn
n
n
nn
nnnn
nn
nn
n
n
n
n
v
v
v
v
K
KK
KKK
KKKK
v
v
v
v
C
CC
CCC
CCCC
v
v
v
v
M
M
M
M
1
2
1
111
21222
1111211
1
2
1
111
21222
1111211
1
2
1
1
2
1
.
.
...
....
..
..
.
.
...
....
..
..
.
.
0
...
....
00..
00..0
tg
MM
M
M
n
n
n
n
1
2
1
1
2
1
.
.
0
...
....
00..
00..0
Symmetric
Symmetric
Symmetric
Sym
metric
Actually, in general 3-D analysis, each element of the above matrices could be a6x6 matrix.
Farzad Naeim Structural Dynamics for Practicing Engineers 6of 71
(Last Revision Date: 5-18-2009)
MDOF SYSTEMS:
ORTHOGONALITY OF MODES
Bettisreciprocal work theorem can be used to develop two orthogonalityproperties of vibration mode shapes
and
It is further assumed for convenience that
)(}0{}]{[}{ nmM mT
n
)(}0{}]{[}{ nmK mT
n
)(}0{}]{[}{ nmC mT
n
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Farzad Naeim Structural Dynamics for Practicing Engineers 7of 71
(Last Revision Date: 5-18-2009)
As we will see, orthogonalityreduces:
n
n
nn
nnnn
nn
nn
n
n
nn
nnnn
nn
nn
n
n
n
n
v
v
vv
K
KK
KKKKKKK
v
v
vv
C
CC
CCCCCCC
v
v
vv
M
M
MM
1
2
1
111
21222
1111211
1
2
1
111
21222
1111211
1
2
1
1
2
1
.
.
...
....
..
..
.
.
...
....
..
..
.
.
0
...
....
00..00..0
tg
M
M
M
M
n
n
n
n
1
2
1
1
2
1
.
.
0
...
....
00..
00..0
Symmetric
Symmetric
Symmetric
Symmetric
MDOF SYSTEMS:
ORTHOGONALITY OF MODES
to:
n
n
n
n
n
n
n
n
n
n
n
n
v
v
v
v
K
K
K
K
v
v
v
v
C
C
CC
C
v
v
v
v
M
M
M
M
*
*
.
.
*
*
0..00
0..00
......
......
00..
00..0
*
*
.
.
*
*
0..00
0..00
......
......
0..0
00..0
*
*
.
.
*
*
0..00
0..00
......
.....
00..0
00..0
1
2
1
*
*1
*2
*1
1
2
1
*
*1
0*1
*1
1
2
1
*
*1
*2
*1
tg
n
n
L
L
L
L
1
2
1
.
.
ornset of independent equations.
This is a monumental achievement which drastically reduces the necessarycomputational efforts.
Farzad Naeim Structural Dynamics for Practicing Engineers 8of 71
(Last Revision Date: 5-18-2009)
DAMPING IN NONLINEAR ANALYSIS
For linear systems we used modesuperposition method.
In mode superposition method thedamping ratio was defined for eachmode of vibration.
This is not possible for a nonlinearsystem because it has no truevibration modes.
A useful way to define the dampingmatrix for a nonlinear system is toassume that it can be represented asa linear combination of the mass andstiffness matrices of the initialelastic system.
This is called the Rayleighdamping.
][][][ KMC
An Example of Rayleigh Damping Functions
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Farzad Naeim Structural Dynamics for Practicing Engineers 9of 71
(Last Revision Date: 5-18-2009)
DAMPING IN NONLINEAR ANALYSIS
and are scalar multipliers whichmay be selected so as to provide agiven percentage of criticaldamping at any two periods ofvibrations.
If damping at the two selected
periods are1 and2, then:
][][][ KMC
An Example of Rayleigh Damping Functions
2
1
21
22
21
12
12
112
Farzad Naeim Structural Dynamics for Practicing Engineers 10of 71
(Last Revision Date: 5-18-2009)
Basic Idealizations of
Nonlinear Behavior
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Farzad Naeim Structural Dynamics for Practicing Engineers 11of 71
(Last Revision Date: 5-18-2009)
BASIC IDEALIZATIONS OF NONLINEAR BEHAVIOR
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
Farzad Naeim Structural Dynamics for Practicing Engineers 12of 71
(Last Revision Date: 5-18-2009)
TWO TYPES OF STRENGTH DEGRADATION
Strength and stiffness degrading model
-600
-400
-200
0
200
400
600
-400 -300 -200 -100 0 100 200 300 400
Displacement
F
orce
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Farzad Naeim Structural Dynamics for Practicing Engineers 13of 71
(Last Revision Date: 5-18-2009)
MEASURED VERSUS CALCULATED RESPONSE
Comparison with of calculated versus experimental results
(a) a moment-critical column (b) a shear-critical column
Source: Kaul R., and Deierlein, G.G. (2004), Object oriented development of strength and stiffness degrading models for reinforced concrete structures,
Farzad Naeim Structural Dynamics for Practicing Engineers 14of 71
(Last Revision Date: 5-18-2009)
Brute Force
Calculation of
Nonlinear Response
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Farzad Naeim Structural Dynamics for Practicing Engineers 15of 71
(Last Revision Date: 5-18-2009)
RESPONSE OF NONLINEAR SDOF SYSTEMS
)()()()( ttpttfttfttf sdi
1
1
1
)()(
)()()(
)()()()(
)(
)(
n
i
iit
n
i
tiis
d
i
tvtkr
tvttvtv
tvtkrtvtkf
ttvcf
ttvmf
)()()( ttgmttpttp e
For response to ground motions:
For response to ground motions:
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
interstorydrift
baseshear/baseshearatyield
)()()( ttgmvkttvcttvm ii and
Hysteretic Loops
Farzad Naeim Structural Dynamics for Practicing Engineers 16of 71
(Last Revision Date: 5-18-2009)
NUMERICAL INTEGRATION
Various methods exist for integration ofVarious methods exist for integration ofequation of motion, including:equation of motion, including:
The Central Difference MethodThe Central Difference Method
TheThe HouboltHoubolt MethodMethod
TheThe NewmarkNewmark-- MethodsMethods The WilsonThe Wilson-- MethodMethod In each case the timeIn each case the time--step chosen (step chosen (t) must bet) must be
small enough to capture the variation andsmall enough to capture the variation anddetails in input ground motions and hystereticdetails in input ground motions and hysteretic
models.models.
See Bathe,See Bathe, Finite Element Procedures in Engineering Analysis,Finite Element Procedures in Engineering Analysis,Prentice Hall, 1982, for more details.Prentice Hall, 1982, for more details.
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Farzad Naeim Structural Dynamics for Practicing Engineers 17of 71
(Last Revision Date: 5-18-2009)
THE CENTRAL DIFFERENCE METHOD
t
vv
v
ii
i 2
11
112112
12
1
2
11
iiiiiiiii
i vvvtt
vv
t
vv
tt
vv
v
assuming1iv
v
t
iv
1iv
t t
t
vvz iii
11 and using the equation of motion
22
1
1
c
t
mzfp
c
t
mz isii
or: 11 iii ztvv
To start, given the initial conditions, Taylor series is used:
00012
vt
vtvv
For numerical stability:
Tt
10
1
5
1 Much smaller values are used for dynamic response toearthquake ground motions (i.e., 0.005 to 0.02 sec.)
Farzad Naeim Structural Dynamics for Practicing Engineers 18of 71
(Last Revision Date: 5-18-2009)
THE NEWMARK- METHODS
1iv
v
t
iv
1iv
t t 1
22
1
11
1111
2
1
22
iiiii
iiii
iiii
vtvtvtvv
vt
vt
vv
pkvvcvm
(1)
(2)
(3)
Knowing the three unknowns:are found by solving the above three simultaneousequations.
iii vvv ,, 111 ,, iii vvv
4
10
)2.0forstablely(numericalMethodNewmarkExplicit0
)3.0forstablely(numericalMethodGoodman&Fox12
1
)4.0forstablely(numericalnamenobutUsed,8
1)5.0forstablely(numericalMethodonAcceleratiLinear6
1
stable)ynumericall(alwaysMethodonAcceleratiConstant4
1
Tt
Tt
Tt
Tt
Much smaller values
are used for dynamicresponse toearthquake groundmotions (i.e., 0.005to 0.02 sec.)
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Farzad Naeim Structural Dynamics for Practicing Engineers 19of 71
(Last Revision Date: 5-18-2009)
NUMERICAL INTEGRATION EXAMPLE
-1
0
1
0 1 2 3 4 5
Time (sec.)
GroundA
cceleration(m/s/s)
K=1000 kN/m
m =100 kN.s2/m
-0.05, -50
1, 50
1.1, 20
0, 0
0.05, 50
-1.1, -20
-1, -50-60
-40
-20
0
20
40
60
-2 -1 0 1 2
Displacement (m)
F
orce(kN)
Example 1:
The SDOF structure shown is excited by horizontalground acceleration as shown below. Assume 0%damping. Find the maximum force and displacementexperienced by the structure if:
1. The structure is elastic and has an infinite amount ofstrength.
2. The structure has an elastic-plastic force-displacement property as shown below and begins tocollapse when displacement exceeds 1.0 m.
Source: Modified fromArmouti, N. S., Earthquake Engineering, Theory and Implementation,Printed in Jordan, 2004,Pages 113-116.
Farzad Naeim Structural Dynamics for Practicing Engineers 20of 71
(Last Revision Date: 5-18-2009)
NUMERICAL INTEGRATION EXAMPLE (CONTINUED)
Use the Central Difference Method to obtain:
Source: Modified fromArmouti, N. S., Earthquake Engineering, Theory and Implementation,Printed in Jordan, 2004,Pages 113-116.
50.00.04617.80.0183.8019
50.0-0.06416.50.0173.6018
50.0-0.17724.60.0253.4017
50.0-0.28746.90.0473.2016
50.0-0.38582.40.0823.0015
24.0-0.462124.90.1252.8014
-31.3-0.518149.50.1502.6013
-50.0-0.536138.30.1382.4012
-50.0-0.51987.70.0882.2011
-50.0-0.47410.10.0102.0010
-50.0-0.408-71.6-0.0721.809
-50.0-0.331-132.6-0.1331.608
-50.0-0.250-156.6-0.1571.407
-50.0-0.173-142.0-0.1421.206
-50.0-0.107-102.5-0.1031.005
-50.0-0.062-62.1-0.0620.804
-28.8-0.029-28.8-0.0290.603
-8.0-0.008-8.0-0.0080.402
0.00.0000.00.0000.201
0.00.0000.00.0000.000
Force (kN)Inelastic Disp. (m)Force (kN)Elastic Disp. (m)Time (sec.)Step No
Inelastic ResponseElastic Response
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Farzad Naeim Structural Dynamics for Practicing Engineers 21of 71
(Last Revision Date: 5-18-2009)
NUMERICAL INTEGRATION EXAMPLE (CONTINUED)
Source: Modified fromArmouti, N. S., Earthquake Engineering, Theory and Implementation,Printed in Jordan, 2004,Pages 113-116.
-22.80.26011.80.0126.2031
-45.50.23819.30.0196.0030
-50.00.23319.00.0195.8029
-35.30.24911.20.0115.6028
-5.70.278-1.1-0.0015.4027
26.20.310-13.0-0.0135.2026
47.70.332-19.7-0.0205.0025
50.00.334-18.5-0.0184.8024
50.00.316-9.9-0.0104.6023
50.00.2792.70.0034.4022
50.00.22114.10.0144.2021
50.00.14420.00.0204.0020
Force (kN)Inelastic Disp. (m)Force (kN)Elastic Disp. (m)Time (sec.)Step No
Inelastic ResponseElastic Response
Farzad Naeim Structural Dynamics for Practicing Engineers 22of 71
(Last Revision Date: 5-18-2009)
NUMERICAL INTEGRATION EXAMPLE (CONTINUED)
-200
-100
0
100
200
-1.0 -0.5 0.0 0.5 1.0
Displacement (m)
RestoringF
orce(kN)
Elastic Force (kN)
Inelastic Force (kN)
Source: Modified fromArmouti, N. S., Earthquake Engineering, Theory and Implementation,Printed in Jordan, 2004,Pages 113-116.
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Time (sec.)
Displacement(m)
Inelastic Disp. (m)
Elastic Disp. (m)
Sd
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Farzad Naeim Structural Dynamics for Practicing Engineers 23of 71
(Last Revision Date: 5-18-2009)
The Concepts of
Nonlinear Response
and Design Spectra
Farzad Naeim Structural Dynamics for Practicing Engineers 24of 71
(Last Revision Date: 5-18-2009)
Elastoplastic Force-Deformation Relation
yy u
u
f
fR 00
Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Farzad Naeim Structural Dynamics for Practicing Engineers 34of 71
(Last Revision Date: 5-18-2009)
INELASTIC DESIGN SPECTRA
Have you ever seen or used aHave you ever seen or used a
nonlinear or inelastic responsenonlinear or inelastic response
spectrum before?spectrum before?
NLSPECTRUM
YES, YOU HAVE.YES, YOU HAVE.
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Source: Chopra, A., Earthquake Dynamics of Structures A Primer, 2nd Edition, EERI, 2005.
Farzad Naeim Structural Dynamics for Practicing Engineers 36of 71
(Last Revision Date: 5-18-2009)
The Concept of
Equivalent
Linearization
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Farzad Naeim Structural Dynamics for Practicing Engineers 37of 71
(Last Revision Date: 5-18-2009)
THE CONCEPT OF EQUIVALENT LINEARIZATION
Consider Free Vibration of a Linear System:
k1
v
fs + fd
A-A
0.1
sin1coscos
sinsinsin
22
2
2
2
A
v
Ac
f
tActAcvcftAv
A
vttkAkvftAv
d
d
s
v
fd
A-A
+
k1
v
fs
A-A
Farzad Naeim Structural Dynamics for Practicing Engineers 38of 71
(Last Revision Date: 5-18-2009)
THE CONCEPT OF EQUIVALENT LINEARIZATION
ntDisplacemeElasticMaximum
ForceElasticMaximum
max, 2 AfAreaDE s
cAAcAArea 2
Now Consider Free Vibration of a Simple
Nonlinear System:
keq1
v
F
A-A k0
Actual
Equivalent
Area of the ellipse Energy Dissipated in one Cycle
kmcm
k
m
c
2;
2
MAXMAX EE
eqF
ED
2
1
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Farzad Naeim Structural Dynamics for Practicing Engineers 39of 71
(Last Revision Date: 5-18-2009)
BASIC LINEARIZATION EXAMPLE
K
m =1000 kN.s2/mExample 2:
The SDOF structure shown below is subjected to horizontalground excitation represented by the pseudo-velocity response
spectrum shown. Stiffness and strength properties of the systemare shown on the hysteresis loop below. Assume zero systemviscous damping and calculate the following.
1. The elastic strength demand and the correspondingdisplacement.
2. The maximum inelastic displacement demand using basic
equivalent linearization technique.
Source: Modified fromArmouti, N. S., Earthquake Engineering, Theory and Implementation,Printed in Jordan, 2004,Pages 117-122.
0.0
0.1
0.20.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
Period (sec.)
SV(m/sec.)
"0% Damping"
"25% Damping"
"50% Damping"FORCE
DISP.
Fy= 1600 kN.
y= +20 mm
Farzad Naeim Structural Dynamics for Practicing Engineers 40of 71
(Last Revision Date: 5-18-2009)
EXAMPLE 2 (CONTINUED)
rad/s8.941000
000,800
m
k
m/s56.00;sec.7.02
vST
ELASTIC RESPONSE
kN/m000,8002.0
16000
y
yFk
kN006,5006.5000,1
m062.0
m/s006.556.094.8S
m062.094.8
56.0
2
a
aE
dE
v
vd
mSF
S
S
SS
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Farzad Naeim Structural Dynamics for Practicing Engineers 41of 71
(Last Revision Date: 5-18-2009)
EXAMPLE 2 (CONTINUED)
m062.0:Assume max E
%43
062.01600
02.02062.0216002
2
1222
2
1
2
1
sec.24.12
rad/s08.5000,1
806,25
kN/m806,25062.0
1600
max
max
MAXMAXMAXMAX EE
yy
EE
eq
eq
eq
eq
eq
y
eq
F
F
F
ED
T
m
K
FK
INELASTIC RESPONSE
ITERATION 1:
m/s45.0%43;sec.24.1 vST
needediterationAnotherm062.0m089.008.5
45.0
vd
SS
Farzad Naeim Structural Dynamics for Practicing Engineers 42of 71
(Last Revision Date: 5-18-2009)
EXAMPLE 2 (CONTINUED)
m/s6.14.04 2aS yad
FmSF
S
kN600,16.1000,1
m1.0
max
max
m10.0:Assume max E
%51
1.01600
02.021.0216002
2
1222
2
1
2
1
sec.57.12
rad/s4000,1
000,16
kN/m000,1610.0
1600
max
max
MAXMAXMAXMAX EE
yy
EE
eq
eq
eq
eq
eq
y
eq
F
F
F
ED
T
m
K
FK
INELASTIC RESPONSE
ITERATION 2:
m/s4.0%51;sec.57.1 vST
acheived.eConvergencm1.0m1.04
4.0
vd
SS
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Farzad Naeim Structural Dynamics for Practicing Engineers 43of 71
(Last Revision Date: 5-18-2009)
The Concept of
Nonlinear Static or
Push-Over Analysis
Farzad Naeim Structural Dynamics for Practicing Engineers 44of 71
(Last Revision Date: 5-18-2009)
DESIGN SPECTRA REPRESENTATIONS
Ordinary DesignOrdinary Design
Period
V/W(Acceleration)
DESIGN SPECTRUM
Spectral or Roof-top Displacement
V/W(Acce
leration)
ConstantPeriodLines
ELASTIC DEMAND SPECTRUM
PushPush--Over AnalysisOver Analysis
Composite or ADRSComposite or ADRSPlotPlot
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Farzad Naeim Structural Dynamics for Practicing Engineers 45of 71
(Last Revision Date: 5-18-2009)
PUSH-OVER CURVE OR CAPACITY SPECTRUM
Using simple modal analysis
equations, spectraldisplacement and roof-topdisplacement may be convertedto each other.
Roof-top Displacement
V/W(Acceleration)
Low-Strength; Low-Stiffness; Brittle
Moderate Strength and Stiffness; Ductile
High-Strength; High-Stiffness; Brittle
Farzad Naeim Structural Dynamics for Practicing Engineers 46of 71
(Last Revision Date: 5-18-2009)
ATC-40 CAPACITY SPECTRUM METHOD
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Farzad Naeim Structural Dynamics for Practicing Engineers 47of 71
(Last Revision Date: 5-18-2009)
ASCE-41 COEFFICIENTS METHOD
Farzad Naeim Structural Dynamics for Practicing Engineers 48of 71
(Last Revision Date: 5-18-2009)
NSP OR PUSH-OVER ANALYSIS
V/W(Acceleration)
Roof-top Displacement
5% damped elastic spectrum
This is an iterative procedure involving severalThis is an iterative procedure involving several
analyses.analyses.
e
For each analysis an effective period for anFor each analysis an effective period for an
equivalent elastic system and aequivalent elastic system and a
corresponding elastic displacement arecorresponding elastic displacement are
calculated.calculated.
This displacement is then divided by a dampingThis displacement is then divided by a damping
factor to obtain an estimate of real displacementfactor to obtain an estimate of real displacement
at that step of analysis.at that step of analysis.
Teff
T0
e/B
ATCATC--4040
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Farzad Naeim Structural Dynamics for Practicing Engineers 49of 71
(Last Revision Date: 5-18-2009)
NSP OR PUSH-OVER ANALYSIS
V/W(Acceleration)
Roof-top Displacement
5% damped elastic spectrum
capacity spectrum
e
Here an estimate of elastic displacementHere an estimate of elastic displacement
is obtained first.is obtained first.
This displacement is then multiplied by aThis displacement is then multiplied by a
set of modification factors to arrive at anset of modification factors to arrive at an
estimate of the target inelasticestimate of the target inelastic
displacement.displacement.
ASCE 41ASCE 41
Farzad Naeim Structural Dynamics for Practicing Engineers 50of 71
(Last Revision Date: 5-18-2009)
ATC-40 CAPACITY SPECTRUM METHOD
This is inherently an iterative procedure.
65.1
100ln41.031.2
12.2
100ln68.021.3
1
11205.0
10
eff
V
eff
A
effeq
eq
SR
SR
TT
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Farzad Naeim Structural Dynamics for Practicing Engineers 51of 71
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MODIFIED CAPACITY SPECTRUM METHOD PER FEMA-440
This is inherently an iterative procedure.
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
Farzad Naeim Structural Dynamics for Practicing Engineers 52of 71
(Last Revision Date: 5-18-2009)
MODIFIED CAPACITY SPECTRUM METHOD PER FEMA-440
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
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Farzad Naeim Structural Dynamics for Practicing Engineers 53of 71
(Last Revision Date: 5-18-2009)
MODIFIED CAPACITY SPECTRUM METHOD PER FEMA-440
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
Farzad Naeim Structural Dynamics for Practicing Engineers 54of 71
(Last Revision Date: 5-18-2009)
MODIFIED CAPACITY SPECTRUM METHOD PER FEMA-440
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
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Farzad Naeim Structural Dynamics for Practicing Engineers 55of 71
(Last Revision Date: 5-18-2009)
COEFFICIENTS METHOD AS MODIFIED PER FEMA-440
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
MODIFIEDC1COEFFICIENT: g
TSCCCC e
at 2
2
3210 4
Farzad Naeim Structural Dynamics for Practicing Engineers 56of 71
(Last Revision Date: 5-18-2009)
COEFFICIENTS METHOD AS MODIFIED PER FEMA-440
gT
SCCCC eat 2
2
32104
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
MODIFIEDC1COEFFICIENT:
y
a
F
mSR
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Farzad Naeim Structural Dynamics for Practicing Engineers 57of 71
(Last Revision Date: 5-18-2009)
COEFFICIENTS METHOD AS MODIFIED PER FEMA-440
gT
SCCCC eat 2
2
3210 4
Source: FEMA 440: Improvement of Nonlinear Static Seismic Analysis Procedures, Feb., 2005.
C3COEFFICIENT ELIMINATED AND REPLACED
WITH A MINIMUM STRENGTH REQUIREMENT: X
motionsgroundfield-nearfor8.0
motionsgroundfield-farfor2.0
ln15.01
4
2
max
PPe
t
e
y
d
y
a
Tt
RF
mSR
Farzad Naeim Structural Dynamics for Practicing Engineers 58of 71
(Last Revision Date: 5-18-2009)
NSP OR PUSH-OVER ANALYSIS
Push-over analysis is in reality the extension of responsespectrum analysis in order to perform approximatenonlinear analysis.
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Farzad Naeim Structural Dynamics for Practicing Engineers 59of 71
(Last Revision Date: 5-18-2009)
INCLUSION OF P-DELTA EFFECTS AND GRAVITY FRAMING ARE VITAL!
ROOF DRIFT ANGLE vs. NORMAL IZED BASE SHEAR
Pushover: LA 20-Story, Pre-Northridge, Model M2, =0%, 3%, 5%, 10%
0.00
0.05
0.10
0.15
0.20
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Roof Drift Angle
NormalizedBaseShear(V/W)
Strain-Hardening =0%
Strain-Hardening =3%
Strain-Hardening =5%
Strain-Hardening =10%
Source: Krawinkler, H. (2005), A few comments on P-Delta, Overstrength, Drift, Deformation Capacity, Collapse Capacity, and other issues,
Presentation at the LATBSDC Invitational Workshop, September 22, Los Angeles.
Typical push-over analysis curves when P- effects are properly considered
Farzad Naeim Structural Dynamics for Practicing Engineers 60of 71
(Last Revision Date: 5-18-2009)
Modeling Nonlinear
Behavior for Dynamic
Analysis
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Farzad Naeim Structural Dynamics for Practicing Engineers 61of 71
(Last Revision Date: 5-18-2009)BASIC INGREDIENTS
Nonlinear material model for theNonlinear material model for theelements and connections must beelements and connections must be
defineddefined
Backbone CurvesBackbone Curves
Hysteretic BehaviorHysteretic Behavior
This could be done usingThis could be done using
Results of experimental researchResults of experimental research
Publications such as FEMAPublications such as FEMA--356, ATC356, ATC--6262
Journal papers and proceedings ofJournal papers and proceedings ofTechnical SeminarsTechnical Seminars
Farzad Naeim Structural Dynamics for Practicing Engineers 62of 71
(Last Revision Date: 5-18-2009)
BACKBONE CURVE EXAMPLE
Source: British Columbia Schools Retrofit Project, Draft Guideline, 2006.
Backbone curve for blocked OSB/plywood shear wall system
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Farzad Naeim Structural Dynamics for Practicing Engineers 63of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: Naeim, Mehrain and Alimoradi, External Peer Review of British Columbia Schools Retrofit Project, Draft Guideline, 2006.
The effect of strengthdegradation on theresponse of the first
floor is evident.
Farzad Naeim Structural Dynamics for Practicing Engineers 64of 71
(Last Revision Date: 5-18-2009)
BACKBONE CURVE AND HYSTERETIC LOOPS EXAMPLE
Source: British Columbia Schools Retrofit Project, Draft Guideline, 2006.
Backbone curve for a tension-only CBF
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Farzad Naeim Structural Dynamics for Practicing Engineers 65of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: Naeim, Mehrain and Alimoradi, External Peer Review of British Columbia Schools Retrofit Project, Draft Guideline, 2006.
The effect of strengthdegradation on theresponse of the first
floor is evident.
Farzad Naeim Structural Dynamics for Practicing Engineers 66of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: ATC-62 Draft, 2006.
Springs 1a & b: Gravity system in buildings
Figure 1. Backbone and hysteretic behavior of Spring 1a.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Interstory Drift Ratio
F / Fy
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
interstorydrift
baseshear/baseshearatyield
Figure 2. Backbone and hysteretic behavior of Spring 1b.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Interstory Drift Ratio
F / Fy
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
interstorydrift
baseshear/baseshearatyield
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Farzad Naeim Structural Dynamics for Practicing Engineers 67of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: ATC-62 Draft, 2006.
Figure 5. Backbone and hysteretic behavior of Spring 2b.
Spring 2a & 2b: Non-ductile moment resisting frame
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
InterstoryDrift Ratio
F / Fy
Figure 4. Backbone and hysteretic behavior of Spring 2a.
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/baseshearatyield
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
InterstoryDrift Ratio
F / Fy
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/baseshearatyield
Farzad Naeim Structural Dynamics for Practicing Engineers 68of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: ATC-62 Draft, 2006.
Spring 3a & 3b: Ductile moment resisting frame
Figure 7. Backbone and hysteretic behavior of Spring 3a.
Figure 8. Backbone and hysteretic behavior of Spring 3b.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
interstorydrift
baseshear/baseshearatyield
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/b
aseshearatyield
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Farzad Naeim Structural Dynamics for Practicing Engineers 69of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: ATC-62 Draft, 2006.
Spring 4a & 4b: Nonductilebrace frames
Figure 10. Backbone and hysteretic behavior of Spring 4a.
Figure 11. Backbone and hysteretic behavior of Spring 4b.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/baseshearatyield
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1.5
-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/baseshearatyield
Farzad Naeim Structural Dynamics for Practicing Engineers 70of 71
(Last Revision Date: 5-18-2009)
HYSTERETIC PROPERTIES EXAMPLE
Source: ATC-62 Draft, 2006.
Spring 5a & b : Infill Walls
Figure 13. Backbone and hysteretic behavior of Spring 5a.
Figure 14. Backbone and hysteretic behavior of Spring 5b.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.5
1
1.5
interstorydrift
baseshear/baseshearatyield
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1.5
-1
-0.5
0
0.5
1
1.5ForcevsDisp
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.5
1
1.5
interstorydrift
baseshear/bas
eshearatyield
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08-1.5
-1
-0.5
0
0.5
1
1.5
interstorydrift
baseshear/base
shearatyield
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Time for a break!
Thank You.Thank You.