ashraf pbd seminar
TRANSCRIPT
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Nonlinear Analysis & Performance Based Design
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Nonlinear Analysis & Performance Based Design
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Nonlinear Analysis & Performance Based Design
CALCULATED vs MEASURED
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Nonlinear Analysis & Performance Based Design
ENERGY DISSIPATION
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Nonlinear Analysis & Performance Based Design
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Nonlinear Analysis & Performance Based Design
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Nonlinear Analysis & Performance Based Design
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Nonlinear Analysis & Performance Based Design
IMPLICIT NONLINEAR BEHAVIOR
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Nonlinear Analysis & Performance Based Design
STEEL STRESS STRAIN RELATIONSHIPS
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Nonlinear Analysis & Performance Based Design
INELASTIC WORK DONE!
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Nonlinear Analysis & Performance Based Design
CAPACITY DESIGN
STRONG COLUMNS & WEAK BEAMS IN FRAMES
REDUCED BEAM SECTIONS
LINK BEAMS IN ECCENTRICALLY BRACED FRAMES
BUCKLING RESISTANT BRACES AS FUSES
RUBBER-LEAD BASE ISOLATORS
HINGED BRIDGE COLUMNSHINGES AT THE BASE LEVEL OF SHEAR WALLS
ROCKING FOUNDATIONS
OVERDESIGNED COUPLING BEAMS
OTHER SACRIFICIAL ELEMENTS
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Nonlinear Analysis & Performance Based Design
MOMENT ROTATION RELATIONSHIP
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Nonlinear Analysis & Performance Based Design
IDEALIZED MOMENT ROTATION
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Nonlinear Analysis & Performance Based Design
PERFORMANCE LEVELS
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Nonlinear Analysis & Performance Based Design
IDEALIZED FORCE DEFORMATION CURVE
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Nonlinear Analysis & Performance Based Design 17
ASCE 41 BEAM MODEL
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Nonlinear Analysis & Performance Based Design
Linear Static Analysis
Linear Dynamic Analysis(Response Spectrum or Time History Analysis)
Nonlinear Static Analysis(Pushover Analysis)
Nonlinear Dynamic Time History Analysis
(NDI or FNA)
ASCE 41 ASSESSMENT OPTIONS
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Nonlinear Analysis & Performance Based Design
ELASTIC STRENGTH DESIGN - KEY STEPS
CHOSE DESIGN CODE AND EARTHQUAKE LOADS
DESIGN CHECK PARAMETERS STRESS/BEAM MOMENT
GET ALLOWABLE STRESSES/ULTIMATEPHI FACTORS
CALCULATE STRESSESLOAD FACTORS (ST RS TH)
CALCULATE STRESS RATIOS
INELASTIC DEFORMATION BASED DESIGN -- KEY STEPS
CHOSE PERFORMANCE LEVEL AND DESIGN LOADSASCE 41
DEMAND CAPACITY MEASURESDRIFT/HINGE ROTATION/SHEARGET DEFORMATION AND FORCE CAPACITIES
CALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)
CALCULATE D/C RATIOSLIMIT STATES
STRENGTH vs DEFORMATION
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Nonlinear Analysis & Performance Based Design
STRUCTURE and MEMBERS
For a structure, F = load, D = deflection.
For a component, F depends on the component type, D is the
corresponding deformation.
The component F-D relationships must be known.
The structure F-D relationship is obtained by structural analysis.
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Nonlinear Analysis & Performance Based Design
FRAME COMPONENTS
For each component type we need :
Reasonably accurate nonlinear F-D relationships.
Reasonable deformation and/or strength capacities. We must choose realistic demand-capacity measures, and it must be
feasible to calculate the demand values.
The best model is the simplest model that will do the job.
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Nonlinear Analysis & Performance Based Design
Force (F)
Initiallylinear
StrainHardening
Ultimatestrength
Strength loss
Residual strength
Complete failure
First yield
Ductile limit
Hysteresis loop
Stiffness, strength and ductile limit mayall degrade under cyclic deformation
Deformation (D)
F-D RELATIONSHIP
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Nonlinear Analysis & Performance Based Design
LATERALLOAD
Partially DuctileBrittle Ductile
DRIFT
DUCTILITY
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Nonlinear Analysis & Performance Based Design
ASCE 41 - DUCTILE AND BRITTLE
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Nonlinear Analysis & Performance Based Design
FORCE AND DEFORMATION CONTROL
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Nonlinear Analysis & Performance Based Design
F
Relationship allowingfor cyclic deformation
ono on c - re a ons p
Hysteresis loops
from experiment.
D
BACKBONE CURVE
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Nonlinear Analysis & Performance Based Design
HYSTERESIS LOOP MODELS
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Nonlinear Analysis & Performance Based Design
STRENGTH DEGRADATION
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Nonlinear Analysis & Performance Based Design
ASCE 41 DEFORMATION CAPACITIES
This can be used for components of all types.
It can be used if experimental results are available.
ASCE 41 gives capacities for many different components.
For beams and columns, ASCE 41 gives capacities only for the chordrotation model.
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Nonlinear Analysis & Performance Based Design
i j
Current
Elastic
Plastic (hinge)
Total
rotation
Zero length hinges
Elastic beam
Plasticrotation
Elasticrotation
Similar atthis end
Mi Mj
M or Mi j
M
i jor
BEAM END ROTATION MODEL
PLASTIC HINGE MODEL
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Nonlinear Analysis & Performance Based Design
PLASTIC HINGE MODEL
It is assumed that all inelastic deformation is concentrated in zero-
length plastic hinges.
The deformation measure for D/C is hinge rotation.
PLASTIC ZONE MODEL
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Nonlinear Analysis & Performance Based Design
PLASTIC ZONE MODEL
The inelastic behavior occurs in finite length plastic zones.
Actual plastic zones usually change length, but models that have
variable lengths are too complex.
The deformation measure for D/C can be :
- Average curvature in plastic zone.- Rotation over plastic zone ( = average curvature x plastic
zone length).
ASCE 41 CHORD ROTATION CAPACITIES
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Nonlinear Analysis & Performance Based Design
ASCE 41 CHORD ROTATION CAPACITIES
IO LS CP
Steel Beam p/y= 1 p/y= 6 p/y= 8
RC Beam
Low shear
High shear
p= 0.01
p= 0.005
p= 0.02
p= 0.01
p= 0.025
p= 0.02
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Nonlinear Analysis & Performance Based Design
COLUMN AXIAL-BENDING MODEL
Mi
PP
P
or
V
V
Mj
MM
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Nonlinear Analysis & Performance Based Design
STEEL COLUMN AXIAL-BENDING
CONCRETE COLUMN AXIAL BENDING
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Nonlinear Analysis & Performance Based Design
CONCRETE COLUMN AXIAL-BENDING
SHEAR HINGE MODEL
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Nonlinear Analysis & Performance Based Design
NodeZero-lengthshear "hinge"Elastic beam
SHEAR HINGE MODEL
PANEL ZONE ELEMENT
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Nonlinear Analysis & Performance Based Design
PANEL ZONE ELEMENT
Deformation, D = spring rotation = shear strain in panel zone.
Force, F = moment in spring = moment transferred from beam to
column elements.
Also, F = (panel zone horizontal shear force) x (beam depth).
BUCKLING RESTRAINED BRACE
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Nonlinear Analysis & Performance Based Design
BUCKLING-RESTRAINED BRACE
The BRB element includes isotropic hardening.
The D-C measure is axial deformation.
BEAM/COLUMN FIBER MODEL
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Nonlinear Analysis & Performance Based Design
BEAM/COLUMN FIBER MODEL
The cross section is represented by a number of uni-axial fibers.
The M-yrelationship follows from the fiber properties, areas and locations.Stress-strain relationships reflect the effects of confinement
WALL FIBER MODEL
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Nonlinear Analysis & Performance Based Design
Steelfibers
Concretefibers
WALL FIBER MODEL
ALTERNATIVE MEASURE STRAIN
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Nonlinear Analysis & Performance Based Design
ALTERNATIVE MEASURE - STRAIN
NONLINEAR SOLUTION SCHEMES
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Nonlinear Analysis & Performance Based Design
u u
1 2iteration
NEWTONRAPHSON
ITERATION
u u
1 2iteration
CONSTANT STIFFNESS
ITERATION
3 4 56
NONLINEAR SOLUTION SCHEMES
CIRCULAR FREQUENCY
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Y
+Y
-Y
0 Radius, R
CIRCULAR FREQUENCY
THE D V & A RELATIONSHIP
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u.
t
ut1.
..
Slope = ut1..
t1
u
t1 t
Slope = ut1.
u
ut1
..
t1 t
THE D, V & A RELATIONSHIP
UNDAMPED FREE VIBRATION
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UNDAMPED FREE VIBRATION
)cos(0 tuut w=
0ukum&&
m
k
where =w
RESPONSE MAXIMA
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Nonlinear Analysis & Performance Based Design
RESPONSE MAXIMA
ut )cos(0 tu w=
)cos(02
tu ww-=ut&&
)sin(0t
u ww-=
ut
&
max2uw-=maxu&&
BASIC DYNAMICS WITH DAMPING
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Nonlinear Analysis & Performance Based Design
BASIC DYNAMICS WITH DAMPING
guuuu &&&&& -22
g
uMKuuCuM
KuuCt
uM
&&&&&
&&&
-
= 0
gu&&
M
K
C
RESPONSE FROM GROUND MOTION
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ug..
2ug2..
ug1..
1
t1 t2
Time t
&&A Bt ug= + =-&& &u u u+ +2
2xw w
RESPONSE FROM GROUND MOTION
DAMPED RESPONSE
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{[& ]cos
[ (& )]sin }
e uB
t
A u uB
tB
t
t d
dt t d
&ut
= -
+ - - + +
-xw
ww
ww xw
ww
w
1 2
2
1 1 2 2
1
e A B t
u uA B
t
A B Bt
t t d
dt t d
ut = - +
+ + - +-
+ - +
-xw
wx
ww
wxw
x
w
x
ww
w
x
w w
{[u ]cos
[&( )
]sin }
[ ]
1 2 3
1 1
2
2
2 3 2
2
1 2 1
2
DAMPED RESPONSE
SDOF DAMPED RESPONSE
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Nonlinear Analysis & Performance Based Design
SDOF DAMPED RESPONSE
RESPONSE SPECTRUM GENERATION
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RESPONSE SPECTRUM GENERATION
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00
TIME, SECONDS
GROUND
ACC,g
Earthquake Record
DisplacementResponse Spectrum
5% damping
-4.00
-2.00
0.00
2.00
4.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DISPL,in.
-8.00
-4.00
0.00
4.008.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DISPL,in.
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10
PERIOD,Seconds
DISPLA
CEMENT,inches
T= 0.6 sec
T= 2.0 sec
SPECTRAL PARAMETERS
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0
4
8
12
16
0 2 4 6 8 10
PERIOD, sec
DISPLACEMENT
,in.
0
10
20
30
40
0 2 4 6 8 10
PERIOD, sec
VELOCITY,in/sec
0.00
0.20
0.40
0.60
0.80
1.00
0 2 4 6 8 10
PERIOD, sec
ACCE
LERATION,g
dV SPS w
va PSPS w
SPECTRAL PARAMETERS
THE ADRS SPECTRUM
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THE ADRS SPECTRUM
ADRS Curve
Spectral Displacement,Sd
SpectralAccele
ration,
Sa
Period, T
SpectralAccelerat
ion,
Sa
RS Curve
0.5
Seco
nds
1.0
Seco
nds
2.0
Seco
nds
THE ADRS SPECTRUM
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THE ADRS SPECTRUM
ASCE 7 RESPONSE SPECTRUM
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ASCE 7 RESPONSE SPECTRUM
PUSHOVER
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Nonlinear Analysis & Performance Based Design
PUSHOVER
THE LINEAR PUSHOVER
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Nonlinear Analysis & Performance Based Design
THE LINEAR PUSHOVER
EQUIVALENT LINEARIZATION
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EQUIVALENT LINEARIZATION
How far to push? The Target Point!
DISPLACEMENT MODIFICATION
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DISPLACEMENT MODIFICATION
DISPLACEMENT MODIFICATION
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Nonlinear Analysis & Performance Based Design
Calculating the Target Displacement
DISPLACEMENT MODIFICATION
C0 Relates spectral to roof displacement
C1Modifier for inelastic displacement
C2Modifier for hysteresis loop shape
d= C0C
1C
2S
aT
e
2/ (4p
2)
LOAD CONTROL AND DISPLACEMENT CONTROL
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LOAD CONTROL AND DISPLACEMENT CONTROL
P-DELTA ANALYSIS
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P DELTA ANALYSIS
P-DELTA DEGRADES STRENGTH
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P DELTA DEGRADES STRENGTH
P-DELTA EFFECTS ON F-D CURVES
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Nonlinear Analysis & Performance Based Design
STRUCTURESTRENGTH
The "over-ductility" is reduced,and is more uncertain.
P- effects may reduce the drift at which the"worst" component reaches its ductile limit.
P- effects will reduce the drift atwhich the structure loses strength.
DRIFT
P DELTA EFFECTS ON F D CURVES
THE FAST NONLINEAR ANALYSIS METHOD (FNA)
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Nonlinear Analysis & Performance Based Design
( )
NON LINEAR FRAME AND SHEAR WALL HINGESBASE ISOLATORS (RUBBER & FRICTION)
STRUCTURAL DAMPERS
STRUCTURAL UPLIFT
STRUCTURAL POUNDING
BUCKLING RESTRAINED BRACES
RITZ VECTORS
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FNA KEY POINT
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The Ritz modes generated by thenonlinear deformation loads are used
to modify the basic structural modes
whenever the nonlinear elements go
nonlinear.
ARTIFICIAL EARTHQUAKES
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Nonlinear Analysis & Performance Based Design
CREATING HISTORIES TO MATCH A SPECTRUM
FREQUENCY CONTENTS OF EARTHQUAKES
FOURIERTRANSFORMS
ARTIFICIAL EARTHQUAKES
ARTIFICIAL EARTHQUAKES
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ARTIFICIAL EARTHQUAKES
MESH REFINEMENT
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Nonlinear Analysis & Performance Based Design71
APPROXIMATING BENDING BEHAVIOR
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Nonlinear Analysis & Performance Based Design72
This is for a coupling beam. A slender pier is similar.
ACCURACY OF MESH REFINEMENT
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PIER / SPANDREL MODELS
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Nonlinear Analysis & Performance Based Design74
STRAIN & ROTATION MEASURES
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STRAIN CONCENTRATION STUDY
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Nonlinear Analysis & Performance Based Design76
Compare calculated strains and rotations for the 3 cases.
STRAIN CONCENTRATION STUDY
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Nonlinear Analysis & Performance Based Design77
The strain over the story height is insensitive to the number of elements.
Also, the rotation over the story height is insensitive to the number of elements.
Therefore these are good choices for D/C measures.
No. of
elems
Roof
drift
Strain in bottom
element
Strain over story
height
Rotation over
story height
1 2.32% 2.39% 2.39% 1.99%
2 2.32% 3.66% 2.36% 1.97%
3 2.32% 4.17% 2.35% 1.96%
DISCONTINUOUS SHEAR WALLS
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Nonlinear Analysis & Performance Based Design
PIER AND SPANDREL FIBER MODELS
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Nonlinear Analysis & Performance Based Design79
Vertical and horizontal fiber models
RAYLEIGH DAMPING
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Nonlinear Analysis & Performance Based Design 80
The aM dampers connect the masses to the ground. They exertexternal damping forces. Units of a are 1/T.
ThebK dampers act in parallel with the elements. They exert internaldamping forces. Units of b are T.
The damping matrix is C= aM+ bK.
RAYLEIGH DAMPING
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Nonlinear Analysis & Performance Based Design 81
For linear analysis, the coefficients aand bcan be chosen to give essentiallyconstant damping over a range of mode periods, as shown.
A possible method is as follows :
Choose TB= 0.9 times the first mode period.
Choose TA= 0.2 times the first mode period.
Calculate aand bto give 5% damping at these two values.
The damping ratio is essentially constant in the first few modes.
The damping ratio is higher for the higher modes.
RAYLEIGH DAMPING
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Nonlinear Analysis & Performance Based Design
KuuC&t
uM&& = 0
Kuu&+ = 0M+b
K)
u&&+
CM
u&M
uK = 0
2uu& =2 0 w2 M = C
w2= +b
2
w2=CM 2
= CM
MK
=2
C
MK 2
MK
M+
b
K=
tuM&&
u&& ;
2
MK
RAYLEIGH DAMPING
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Nonlinear Analysis & Performance Based Design
Higher Modes (high w) =b
Lower Modes (low w) = a
To get z from a&bfor anyw M
K=
To geta&b from two values of z1& z
2
; T 2pw
=
z1= a
2w1
+
b w12
z2= a
2w2
+
b w22
Solve fora&b
DAMPING COEFFICIENT FROM HYSTERESIS
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Nonlinear Analysis & Performance Based Design
DAMPING COEFFICIENT FROM HYSTERESIS
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Nonlinear Analysis & Performance Based Design
A BIG THANK YOU!!!
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