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Transcript of Inelastic Behavior
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Inelastic Behavior of
Materials and Structures
Inelastic Behavior 1Revised 3/10/06
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Inelastic Behavior of
Materials and Structures
Illustrate Inelastic Behavior of Materials and Structures
Explain Why Inelastic Response May be Necessary
Explain the Equal Displacement Concept
Introduce the Concept of Inelastic Design Response Spectra
Explain How Inelastic Behavior is Built into ASCE 7-05
Inelastic Behavior 2Revised 2/29/04
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Importance in Relation to ASCE 7-05
Derivation and Explanation of the Response
Reduction FactorR
Derivation and Explanation of the Displacement
Amplification FactorC
d
Derivation and Explanation of the OverstrengthFactor o
Inelastic Behavior 3
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Inelastic Behavior of Structures
From Material
to Cross Section
to Critical Region
to Structure
Inelastic Behavior 4
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Idealized Inelastic Behavior
From Material..
Inelastic Behavior 5
Strain
Stress
y
y u
u
= u
y
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Stress-Strain Relationships for Steel
Inelastic Behavior 6
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Stress-Strain Relationships for Steel
Inelastic Behavior 7
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Unconfined Concrete
Stress-Strain Behavior
Inelastic Behavior 8
0
2000
4000
6000
800010000
12000
14000
16000
18000
20000
0 0.001 0.002 0.003 0.004
Strain, in./in.
Stres
s,psi
4500 psi
8800 psi
13,500 psi
17,500 psi
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Confined Concrete
Stress-Strain Behavior
Inelastic Behavior 9
0
1000
2000
3000
4000
5000
6000
7000
8000
0 0.01 0.02 0.03 0.04
Average strain on 7.9 in. gauge length
Stress,psi
no confinement
4.75 in.
3.5 in.
2.375 in.
1.75 in.
Pitch of in. dia.
spiral
Tests of
6 in. x 12 in.
cylindersHuge increase in Strain Capacity !
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Confinement by Spirals or Hoops
Asp
ds
fyhAsp
fyhAsp
Confinementfrom square
hoop
Forces actingon 1/2 spiral or
circular hoop
Confinementfrom spiral or
circular hoop
Inelastic Behavior 10
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Concrete Confinement
Rectangular hoops
with cross tiesConfinement by
transverse bars
Confinement by
longitudinal bars
Inelastic Behavior 11
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Inelastic Behavior 12
Idealized Stress-Strain Behavior of
Confined Concrete
Kent and Park Model
0
500
1000
1500
20002500
3000
3500
4000
4500
0 0.004 0.008 0.012 0.016
Strain, in./in.
Stress,psi
No Hoops
4 in.
6 in.9 in.
12 in.
Pitch of
in. dia.
spiral
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Inelastic Behavior 13
Unconfined
Confined
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Benefits of Confinement
Spiral Confinement
Virtually NO Confinement
Olive View Hospital, 1971 San Fernando Valley Earthquake
Inelastic Behavior 14
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IDEALIZED INELASTIC BEHAVIOR
To Section..
Inelastic Behavior 15
Strain Stress Moment
y y My
y
Strain Stress Moment
u u Mu
u
ELASTIC
INELASTIC
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IDEALIZED INELASTIC BEHAVIOR
To Section..
Curvature
Inelastic Behavior 16
Moment
My
y u
Mu
=
u
y
NOTE:
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Software for Moment - Curvature Analysis
XTRACT
Inelastic Behavior 17
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IDEALIZED INELASTIC BEHAVIORTo Critical Region and Member..
Inelastic Behavior 18
LP
Curvature
My
Mu
y u
Moment
y
u
Area u
Area y
ELASTIC INELASTIC
IDEALIZED INELASTIC BEHAVIOR
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IDEALIZED INELASTIC BEHAVIORTo Critical Region and Member..
Inelastic Behavior 19
Rotation
Moment
My
y u
Mu
= u
y
NOTE:
Critical Region Behavior of a Steel Girder
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Critical Region Behavior of a Steel Girder
Inelastic Behavior 20
Strength Loss
Due to Flange
Buckling
IDEALIZED INELASTIC BEHAVIOR
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IDEALIZED INELASTIC BEHAVIORTo Structure..
Inelastic Behavior 21
Force
Displacement
= u
y
Note:
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Loss of Ductility through Hierarchy
Strain = 100
Curvature = 12 to 20
Rotation= 8 to 14
Displacement = 4 to 10
Inelastic Behavior 22
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Ductility and Energy Dissipation Capacity
System ductility of 4 to 6 is required foracceptable seismic behavior.
Good hysteretic behavior requiresductile materials. However, ductility in
itself is insufficient to provide acceptableseismic behavior.
Cyclic energy dissipation capacity is abetter indicator of performance.
Inelastic Behavior 23
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Response Under Reversed
Cyclic Loading
(t)
+
Time
Earthquakes impose DEFORMATIONS. Internal Forces Develop
as a result of the deformations.
Inelastic Behavior 24
Laboratory Specimen under Cyclic Deformation Loading
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Laboratory Specimen under Cyclic Deformation Loading
Load Cell
Displacement
Transducer
(Controls )
Hydraulic
Ram
Records (F)
+
Time
1
8
7
6
5
4
3
2
Moment ArmL
Deformations are cyclic,
inelastic, and reversed
Inelastic Behavior 25
Laboratory Specimen under Cyclic Deformation Loading
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Laboratory Specimen under Cyclic Deformation Loading
Measured Hysteretic
Behavior
Force
Deformation
Load Cell
Hydraulic
Ram
Records (F)
Displacement
Transducer
(Controls ) Moment ArmL
Inelastic Behavior 26
Hysteretic Behavior
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Hysteretic Behavior
Deformation Deformation
Force Force
PINCHED
(Good)ROBUST
(Excellent)
Inelastic Behavior 27
Hysteretic Behavior
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Hysteretic Behavior
ForceForce
Inelastic Behavior 28
Deformation
PINCHED (with strength loss)
Poor
Deformation
x
BRITTLE
Unacceptable
Hysteretic Behavior
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Hysteretic Behavior
Deformation Deformation
ForceForceAREA=Energy Dissipated
ROBUST PINCHED (No Strength Loss)
Inelastic Behavior 29
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Ductility and Energy Dissipation Capacity
The structure should be able to sustain severalcycles of inelastic deformation without significant
loss of strength.
Some loss of stiffness is inevitable, but excessive
stiffness loss can lead to collapse.
The more energy dissipated per cycle, withoutexcessive deformation, the better the behaviorof the structure.
Inelastic Behavior 30
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Ductility and Energy Dissipation Capacity
The art of seismic resistant design is in the details.
With good detailing, structures can be designed
for force levels significantly lower than would berequired for elastic response.
Inelastic Behavior 31
Why is Inelastic Response Necessary?
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Why is Inelastic Response Necessary?
Compare the Wind and Seismic Design of a
Simple Building
Building Properties:
Moment Resisting Frames
Density= 8 pcf
Period T = 1.0 sec
Damping= 5%
Soil Site Class B
120
90
62.5
Wind:
100 MPH Exposure C
Earthquake:
Assume SD1=0.48g
Inelastic Behavior 32
Wi d
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Wind:
90
62.5
120
120 mph 3-Sec GustExposure C
Velocity pressure q= 36.8 psfGust factor G = 0.85
Pressure coefficient Cp= 1.3
Load and Directionality Factor for Wind = 1.6x0.85=1.36
Total wind force on 120 face:
VW120= 62.5*120*36.8*0.85*1.3*1.36/1000 = 415 kipsTotal wind force on 90 face:
VW90 = 62.5*90*36.7*0.85*1.3*1.36/1000 = 310 kips
Inelastic Behavior 33
E th k
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Earthquake:
90
62.5Building Weight W=
120*90*62.5*8/1000 = 5400
kips 120
V C WEQ S=
C S
T R IS
D= = =1 0 48
1 0 1 0 1 00480
( / )
.
. ( . / . ).
Total ELASTIC earthquake force (in each direction):
VEQ = 0.480*5400 = 2592 kips
Inelastic Behavior 34
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Comparison: Earthquake vs. Wind
120
2592 6.2415
EQ
W
VV
= =90
2592 8.4310
EQ
W
VV
= =
ELASTIC Earthquake forces 6.2 to 8.4 times wind!
Virtually impossible to obtain economical design
Inelastic Behavior 35
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How to Deal with Huge Earthquake Force?
Isolate structure from ground (Base Isolation) Increase Damping (Passive Energy Dissipation)
Allow Controlled Inelastic Response
Historically, Building Codes use Inelastic Response Procedure.
Inelastic response occurs through structural damage (yielding).
We must control the damage for the method to be successful.
Inelastic Behavior 36
Assume Frame is Designed for Wind
P h A l i P di t St th 500 k
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Inelastic Behavior 37
Force, kips
Displacement, inches
4.0
400
200
1.0 2.0 3.0
600
5.0
Pushover Analysis Predicts Strength = 500 k
Actual
Idealized
How will Frame Respond During
0 4g El Centro Earthquake?
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0.4g El Centro Earthquake?
Force, kips
4.0
600
Inelastic Behavior 38
400
Fy=500 k
K1=550 k/in
K2=11 k/in
200
2.0 3.0 5.01.0
Idealized SDOF Model
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 2 4 6 8 10 12 14 16 18 20
T im e , Secon ds
Acceleration,g
Displacement, in.
El Centro Ground Motion
Response Computed by NONLIN:
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Maximum
Displacement:
Maximum
Shear Force:
Number ofYield Events:
4.79
542 k
15
Inelastic Behavior 39
Response Computed by NONLIN:
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Response Computed by NONLIN:
Yield Displacement = 500/550 = 0.91 inch
Inelastic Behavior 40
= =Maximum Displacement
Yield Displacement
4 79
0 915 26
.
..Ductility Demand
Interim Conclusion (The Good News)
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Interim Conclusion (The Good News)
The frame, designed for a wind force which is 15%of the ELASTIC earthquake force, can survive the
earthquake if:
It has the capability to undergo numerous cycles ofINELASTIC deformation
It has the capability to deform at least 5 to 6times the yield deformation
It suffers no appreciable loss of strength
REQUIRES ADEQUATE DETAILING
Inelastic Behavior 41
Interim Conclusion (The Bad News)
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Interim Conclusion (The Bad News)
As a result of the large displacements associated with
The inelastic deformations, the structure will suffer
considerable structural and nonstructural damage.
This damage must be controlled byadequate detailing and by limiting
structural deformations (drift)
Inelastic Behavior 42
Development of Equal Displacement
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Development of Equal Displacement
Concept of Seismic Resistant Design.
Concept used by:
UBC/IBC
NEHRP
ASCE-7
FEMA 273
In association with Force Based
design concept. Used to predict
design forces and displacements
In association with Static Push Over
Analysis. Used to predict displacementsat various performance points.
Inelastic Behavior 43
Th E l Di l t C t
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The Equal Displacement Concept
The displacement of an inelastic system, with
stiffness K and strength Fy, subjected to a particularground motion, is approximately equal to the displacement
of the same system responding elastically.
(The displacement of a system is independent of the
yield strength of the system)
Inelastic Behavior 44
Repeat Analysis for Various Yield Strengths
(All other parameters stay the same)
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Inelastic Behavior 45
0
1
2
3
4
5
6
7
0 500 1000 1500 2000 2500 3000 3500 4000
Yield Strength (K)
MaximumD
isplacement(in)
(All other parameters stay the same)
0
1
2
3
4
5
6
0 500 1000 1500 2000 2500 3000 3500 4000
Yield Streng th (K)
DuctilityDemand
Avg. = 5.0 in.
Inelastic
Elastic
Elastic = 5.77
Constant Displacement Idealization
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p
of Inelastic Response
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8
Displacement, Inches
Forc
e,
Kips
ACTUAL BEHAVIOR
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8
Displacement, inches
Forc
e,
Kips
IDEALIZED BEHAVIOR
Inelastic Behavior 46
Equal Displacement Idealization
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q p
of Inelastic Response
For design purposes, it may be assumed that
inelastic displacements are equal to thedisplacements that would occur during an
elastic response.
The required force levels under inelastic responseare much less than the force levels required for
elastic response.
Inelastic Behavior 47
Equal Displacement Concept
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0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8
Displacement, inches
Force,
Kips
Elastic
Inelastic
Design for This Force Design for this Displacement
of Inelastic Design
5.77
Inelastic Behavior 48
Equal Displacement Concept
of Inelastic Design
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0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8
Displacement, inches
Force,
Ki
ps
Elastic
Inelastic
du=5.77dy=0.91
Ductility Supply MUST BE > Ductility Demand =
Inelastic Behavior 49
5 77
0 91
6 34.
.
.=
of Inelastic Design
3500 FE
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0
500
1000
1500
2000
2500
3000
0 2 4 6 8
Displacement, inches
Force,
Kips
Using Response Spectrum, Estimate ELASTIC Force Demand FE
FI
dR dI
Estimate Ductility Supply, , and determine INELASTIC ForceDemand FI = FE / Design Structure for FI
Inelastic Behavior 50
Compute Reduced Displacement dR, and multiply by to obtain true
Inelastic Displacement dI.
Check Drift using di
ASCE-7 Approach
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pp
Use basic elastic spectrum, but for strength,
divide all pseudoacceleration values by R,
a response modification factor that accounts for:
Anticipated ductility supply
Overstrength Damping (if different than 5% critical) Past performance of similar systems
Redundancy
Inelastic Behavior 51
Ductility/Overstrength
FIRST SIGNIFICANT YIELD
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FIRST SIGNIFICANT YIELD
First
Significant
Yield
Force
Design
Strength
Inelastic Behavior 52
Displacement
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First Significant Yieldis the level of force which causes complete
plastification of at least the most critical regionof the structure (e.g. formation of the first
plastic hinge)
The Design Strength of a Structure is equal
to the resistance at first significant yield.
Inelastic Behavior 53
Overstrength (1)
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Force
Design
Strength
Inelastic Behavior 54
Displacement
Overstrength (2)
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Force
Design
Strength
Inelastic Behavior 55
Displacement
Overstrength (3)
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Force
ApparentStrength
Over StrengthDesign
Strength
Inelastic Behavior 56
Displacement
Sources of Overstrength
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Sources of Overstrength
Sequential Yielding of Critical Regions
Material Overstrength (Actual vs Specified Yield) Strain Hardening
Capacity Reduction ( ) Factors Member Selection
Inelastic Behavior 57
Definition of Overstrength Factor
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Overstrength Factor =Apparent Strength
Design Strength
Force
Apparent Strength
Over Strength
Design Strength
Displacement
Inelastic Behavior 58
Definition of Ductility Reduction Factor RdForce
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Elastic StrengthDemand
Inelastic Behavior 59
Design Strength
Apparent Strength
DisplacementElastic
Displacement
Demand
Strength
DemandReduction due
To Ducti lity
Over Strength
Definition of Ductility Reduction Factor
Elastic Strength Demand
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Elastic Strength Demand
Ductility Reduction Rd= Apparent Strength
Inelastic Behavior 60
DesignStrength
Apparent
Strength
Force
Displacement
Elastic
StrengthDemand
Elastic
Displacement
Demand
Strength
Demand
Reduction dueTo Ducti lity
Over Strength
Definition of Response Modification
Coefficient R
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Overstrength Factor =Apparent Strength
Design Strength
Ductility Reduction Rd
=Elastic Strength Demand
Apparent Strength
Elastic Strength Demand
Design Strength =RdR
=
Inelastic Behavior 61
Definition of Response Modification
Coefficient R
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Force
Reduced (Design) Strength
R x Design Strength
DisplacementElastic
Displacement
Demand
x Design Strength
ANALYSIS DOMAIN
Inelastic Behavior 62
Definition of Deflection Amplification Factor Coefficient Cd
Elastic Strength
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Elastic Strength
Demand
Inelastic Behavior 63
Design Strength
Apparent Strength
Elastic
Displacement
DemandE
Actual Inelastic
Displacement
DemandI
Computed Design
Displacement
DemandD
ASCE 7 Approach for Displacements
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Determine Design Forces V=CsW, where C
sincludes
Ductility/Overstrength Reduction Factor R.
Distribute forces vertically and horizontally, and
compute displacements using linear elastic analysis.
Multiply computed displacements by Cd
to obtain
estimate of true inelastic response.
Inelastic Behavior 64
EXAMPLES of Design Factors
for Steel Structures
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Inelastic Behavior 65
ASCE 7-05R o Rd Cd
Special Moment Frame 8 3 2.67 5.5
Intermediate Moment Frame 4.5 3 1.50 4.0Ordinary Moment Frame 3.5 3 1.17 3.0
Eccentric Braced Frame 8 2 4.00 4.0
Eccentric Braced Frame (Pinned) 7 2 3.50 4.0
Special Concentric Braced Frame 6 2 3.00 5.0
Ordinary Concentric Braced Frame 3.25 2 1.25 3.25
Not Detailed 3 3 1.00 3.0
Note: Rd
is Ductil ity Demand ONLY IF o is Achieved.
EXAMPLES of Design Factors
for Reinforced Concrete Structures
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o e o ced Co c ete St uctu es
ASCE 7-05
R o Rd Cd
Special Moment Frame 8 3 2.67 5.5Intermediate Moment Frame 5 3 1.67 4.5
Ordinary Moment Frame 3 3 1.00 2.5
Special Reinforced Shear Wall 5 2.5 2.00 5.0Ordinary Reinforced Shear Wall 4 2.5 1.60 4.0
Detailed Plain Concrete Wall 2 2.5 0.80 2.0
Ordinary Plain Concrete Wall 1.5 2.5 0.60 1.5
Note:Rd is Ductility Demand ONLY IF o is Achieved.
Inelastic Behavior 66
ASCE-7 Elastic Spectra as
Adjusted for Ductility and Overstrength
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j y g
Inelastic Behavior 67
C SR I
SDS=/
C S
T R IS
D= 1
( / )0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4
Period, Seconds
Acc
eleration,g
R=1
R=2
R=3
R=4
R=6
R=8
SDS=1.0g SD1=0.48g
1 2
Using Modified ASCE-7 Spectrum
to Determine Force Demand
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Inelastic Behavior 68
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 1.0 2.0 3.0 4.0 5.0
Period, Seconds
Pseudoacceleration,g
R=1
R=4
0.15V=0.15W
Using Modified ASCE-7 Spectrum
to Determine Displacement Demand
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Inelastic Behavior 69
0
5
10
15
20
25
0 1 2 3 4 5
Period, Seconds
Displa
cement,inches.
R=1
R=4
(R=4)*Cd
Cd=3.5
Displacements must be Multiplied by factor CdBecause Displacements Based on Reduced
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Inelastic Behavior 70
Force would be too low INELASTIC d REDUCED ELASTICC=
0
5
10
15
20
25
0 1 2 3 4 5
Period, Seconds
Displacemen
t,inches.
R=1
R=4
(R=4)*Cd
Cd=3.5
inINELASTIC 65.3=
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Equal Displacement approach may not beapplicable at very low Period Values
Inelastic Behavior 71
Equal Energy Concept(Applicable at Low Periods)
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ELASTIC ENERGY
FE
EE
FI
F
F
E
I
y
E F F
F
F
FE E
E
I
y yE
I
= =0 5 0 52
. .
Inelastic Behavior 72
Equal Energy Concept
(Applicable at Low Periods)
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INELASTIC ENERGY
FI
y u
EI
( )5.05.0 == yIyIuII FFFE
Inelastic Behavior 73
Equal Energy Concept
(Applicable at Low Periods)
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Assuming EE= EI :
FE
FI
F
F
E
I= 2 1
Inelastic Behavior 74
100
Elastic:F
Newmark Inelastic Spectrum (for Psuedoacceleration)
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Inelastic Behavior 75
0.1
1
10
0.01 0.1 1 10
Period, Seconds
Pseud
oVelocity,In/Sec
Equal Energy
Elastic:
Inelastic:
EqualDisp.
Transition
EI
FF =
12 =
EI
F
F
Newmarks
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Inelastic Design Response Spectrum
To obtain Inelastic DISPLACEMENTSpectrum, multiply the Spectrum shown in
Previous slide by (for all Periods)
Inelastic Behavior 76
100
Newmarks Inelastic Design Response Spectrumfor Acceleration and Displacement
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Inelastic Behavior 77
0.1
1
10
0.01 0.1 1 10Period, Seconds
PseudoV
elocity,
In/Sec
Disp.
Accel.
At very low periods, ASCE-7 Spectrum does not reduce to
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ground acceleration, so this partially compensates forerror in equal displacement assumption at low Period
Values.
Period, T
Cs
Note: FEMA 273 has explicit modifies for computing Target
at Low Periods
Inelastic Behavior 78