evaluation of nonlinear seismic demands of high-rise rc
TRANSCRIPT
Evaluation of Nonlinear Seismic Demands of High-rise RC Shear Wall
Buildings using the Simplified Analysis Procedures
Fawad A. Najam
st112966
Asian Institute of Technology (AIT), Thailand.
Email: [email protected]
Final Comprehensive Examination
9th October 2017
Introducing AIT SolutionsFinal Comprehensive Examination 2
Future ground
shaking
Structure
Linear/Nonlinear Analysis Model
Characterization of Seismic
Ground Motions
Estimation of Linear/Nonlinear
Seismic Demands
β’ Global-level Responses
β’ Inter-story Responses
β’ Component-level Responses
The Earthquake Problem
Seismic Waves
Soil
Epicenter
Hypocenter
(Focus)
Fault
AttenuationSeismic waves lengthen and diminish in strength
as they travel away from the ruptured fault.
Site ResponseGround motion can
be amplified by soil
Site
Seismic Waves
Epicenter
Hypocenter
(Focus)
Fault
Soil
Site
Site ResponseGround motion can
be amplified by soil
Attenuation
Introducing AIT SolutionsFinal Comprehensive Examination 3
β¦ β¦ β¦ Simplified Estimation of Seismic Demands
Simplified Structural Models
Simplified Loading
Any combination of simplified approaches
Simplified Seismic Analysis
Procedures
Triangular
Modal Expansion
β¦ β¦ β¦
β¦ β¦ β¦
Simplifying the Problem
4
Determination of Nonlinear Seismic Demands
Seismic
Loading
Structural
Models
Detailed 3D
Nonlinear Model
Equivalent MDF
Model
Equivalent SDF
Model
Static Single Lateral
Load Vector
Static Multiple Lateral
Load Vectors
Ground Acceleration
Records
Relative Uncertainty
LowHigh
The NSPs
The Multi-mode
Pushover Analysis
Procedures
The Detailed
NLRHA Procedure
Analysis Procedures
Introducing AIT SolutionsFinal Comprehensive Examination 5
Study Objectives
β’ Real case study buildings (20-, 33-,
and 44-story high)
β’ Different types and levels of input
ground motions
Conception and theoretical formulation
Application and subsequent validation
Practicality, limitations and sources of uncertainty
β’ Practical and convenient applicability
β’ Future improvements
β’ Modal Decomposition
β’ Higher-mode Effects
Introducing AIT SolutionsFinal Comprehensive Examination 6
Basic Tool to Achieve Study Objectives β The UMRHA Procedure
β + +β¦
Mode 1Mode 2
Mode 3
+
A Detailed 3D Elastic
Structural Model
The Classical Modal
Analysis Procedure
The Uncoupled Modal Response
History Analysis (UMRHA)
Procedure
F
D
F
D
F
D
NLNL NL
A Detailed 3D Inelastic
Structural Model
β
+ + β¦
Mode 1Mode 2
Mode 3
+
Introducing AIT SolutionsFinal Comprehensive Examination 7
Uncoupled Modal Response
History Analysis (UMRHA)
Three case study high-
rise buildings (20-, 33-,
and 44-story high)
Nonlinear Response History
Analysis (NLRHA)
Analysis Scheme to Achieve the Objectives
Four sets of ground
motions with different
levels and characteristics
Individual Modal Responses
Proposed Simplified Analysis
Procedures
Combined Response
Combined Response
Combined Response
Individual Modal Responses
Introducing AIT SolutionsFinal Comprehensive Examination 8
Case study Buildings
44-story
B3
33-story
B2
20-story
B1
β’ Located in Bangkok, Thailand
β’ Heights vary from 20 to 44 stories
β’ RC slab-column frames carry gravity loads
β’ RC walls & cores resist lateral loads
β’ Masonry infill walls extensively used
β’ Designed for wind loads, but not for seismic
effects
β’ Possess irregular features commonly found
in typical tall buildings, e.g. podium and non-
symmetrical arrangement of RC walls, etc.
Introducing AIT SolutionsFinal Comprehensive Examination 9
Nonlinear Modeling of Case Study Buildings
MVLEM for RC Walls Lumped Fibers for RC Columns Masonry Infill Wall Model
Linear-elastic frame
element (shear
deformation excluded)
Fiber section
(concrete and
steel fibers)
Linear shear spring
Equivalent concrete strutsLevel m
Level m-1
Rigid Link
Concrete and
Steel fibers
Linear shear
spring
πΈπ πΈπ πΈπ
Stress
Strainνπβ² νππ’
ππβ²
ππ‘
νπ‘
Mander Envelope
Perform 3D Envelope
Unloading
Reloading
πΈπ
Stress
Strainνπ¦ νπ’
ππ’
ππ¦
νπ β
Park Envelope
Perform 3D
πππΈπ
πΌπΈπ
Concrete
Steel
Introducing AIT SolutionsFinal Comprehensive Examination 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
7.5%-damped
Mean Spectrum
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
2.5%-damped
Mean Spectrum
5%-damped
Mean Spectrum
7.5%-damped
Mean
Spectrum
5%-damped
Mean Spectrum
2.5%-damped
Mean Spectrum
7.5%-damped
Mean Spectrum
5%-damped
Mean Spectrum
2.5%-damped
Mean Spectrum
Set 2Set 1
Sp
ectr
al A
cce
lera
tio
n (
g)
Time Period (sec)
Sp
ectr
al A
cce
lera
tio
n (
g)
Time Period (sec)
Time Period (sec)
Time Period (sec)
7.5%-damped
Mean
Spectrum
5%-damped
Mean Spectrum
2.5%-damped
Mean Spectrum
5%-damped Target Response Spectrum
Mean of Matched Ground Motions
Individual Ground Motions
Set 4Set 3
Final Comprehensive Examination 11
Determination of Nonlinear Seismic Demands β Background
Introducing AIT SolutionsFinal Comprehensive Examination 12
Nonlinear Static Procedures
β’ Several EL Procedures (Individual studies)
β’ Capacity Spectrum Method (CSM) (ATC 40, FEMA 273, FEMA 356)
β’ FEMA 440 Improved EL Procedure
Equivalent Linearization
β’ Several expressions for displacement modification (Individual studies)
β’ Displacement Coefficient Method (ATC 40, FEMA 273, FEMA 356, FEMA 440, ASCE 41-06, ASCE 41-13)
Displacement Modification
Full 3D Nonlinear
MDF Model
πΉπ
πΉ1
.
.
.
Monotonic Pushover
Analysis Procedure
π₯π
ππ
Determination of
Pushover Curve
ππ
π₯π
Idealization of
Pushover Curve
πΉ
π·
An Equivalent
SDF System
The Concept of an βEquivalent SDF Systemβ
Introducing AIT SolutionsFinal Comprehensive Examination 13
Capacity Spectrum Method (ATC 40, 1996)
Full 3D Nonlinear MDF Model
β’ Determine the modal
properties π1, π, Ξ and πΆπ
πΉπ
πΉ1
.
.
.
Monotonic Pushover Analysis
β’ Monotonically push and get
the ππ1 vs. π₯1π relationship
π₯1π
ππ1
ππ1Determination of Responses
π₯1π
β’ Extract the responses from the pushover database at πΏπ
πΉπ
πΉ1
.
.
.
πΏπ
ππ1πΏπ
ππ1
Conversion of Pushover Curve
to ADRS Format
π₯1π
ππ΄ =ππ π πΆπ
ππ΄
ππ·
ππ· =π2
4π2ππ΄
ππ· =π₯π
Ξπ1π
Determination of Performance
Pointππ΄
ππ·
Performance Point
Demand Spectrum
Capacity Spectrum
β’ The βDemand Spectrumβ is the reduced
form of 5% damped spectrum
πΏπ
Introducing AIT SolutionsFinal Comprehensive Examination 14
Displacement Coefficient Method (ATC 40, FEMA 273, FEMA 356)
πΏπ = πΆ0πΆ1πΆ2πΆ3ππ΄ππ2
4π2π
βEffectiveβ periodFor converting spectral
to roof displacement
For converting elastic to
inelastic displacement
To account for the shape
of hysteresis loopsSpectral acceleration at ππ
Target Displacement
To account for dynamic
π β Ξ effects
Introducing AIT SolutionsFinal Comprehensive Examination 15
Improved NSPs β FEMA 440 (2005)
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
Elastic-Perfectly Plastic (EPP)
Fo
rce
Displacement-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
Stiffness-Degrading (SD)
Fo
rce
Displacement
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
Nonlinear Elastic (NE)
Fo
rce
Displacement
EPP SD
SSD NE
Strength and Stiffness-Degrading (SSD)
Fo
rce
Displacement
Introducing AIT SolutionsFinal Comprehensive Examination 16
Inelastic-to-elastic Displacement Ratios (IDRs) (FEMA 440, 2005)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.5 1 1.5 2 2.5 3
π π¦ = 8
π π¦ = 6
π π¦ = 4
π π¦ = 3
π π¦ = 2
π π¦ = 1.5
πΆ1,πΈππ Site Class C β Mean of 20 Ground Motions
Two Basic Spectral Regions
πΆ1 is on average larger than 1
πΆ1 increases with decreasing ππΆ1 increases with increasing π π¦
πΆ1 is approximately constant with changes in ππΆ1 doesnβt change much with changes in π π¦πΆ1 is on average approximately equal to 1
π (sec)
Introducing AIT SolutionsFinal Comprehensive Examination 17
ASCE/SEI 41-13 (2013) Nonlinear Static Procedure
Full 3D Nonlinear MDF Model
β’ Determine the modal
properties π1, π, Ξ and πΆπ
πΉπ
πΉ1
.
.
.
Monotonic Pushover Analysis
β’ Monotonically push and get
the ππ1 vs. π₯1π relationship
π₯1π
ππ1
ππ1
Idealization of Pushover Curve
π₯1π
IdealizedActual
β’ Idealized force-displacement
relationship.
β’ Determine πΎπ, ππ¦ and π π¦
Determination of Target
Displacement
β’ Determine the effective time
period and coefficients
β’ Target displacement
πΏπ = πΆππΆ1πΆ2ππ΄ππ2
4π2π
ππ = πππΎππΎπ
πΆ1 = 1 +π π¦ β 1
πππ2
πΆ2 = 1 +1
800
π π¦ β 1
ππ
2
ππ1Determination of Responses
π₯1π
β’ Extract the responses from the pushover database at πΏπ
πΉπ
πΉ1
.
.
.
πΏπ
ππ1πΏπ
18
Pushover Analysis Procedures
First Modal InertiaSimple (Uniform, Triangular etc.) Multi-mode
Modal Combination of
Loading
Modal Combination of
Response
Consecutive Modal
Pushover (CMP)
Adaptive Modal
Pushover (AMP)
Modal Combination Method (Kalkan and Kunnath, 2004)
Upper-bound Pushover Analysis (Jan et al., 2004)
Multimode Pushover Analysis (Sasaki et al., 1996)
Modal Pushover Analysis (Chopra and Goel, 2002)
Modified Modal Pushover Analysis (Chopra et al., 2004 )
Adaptive Modal Pushover Analysis (Kalkan and Kunnath, 2006)
Optimal Multimode Pushover Analysis (Attard and Fafitis, 2005)
Consecutive Modal Pushover Analysis (Poursha et al., 2009)
Lateral Load Vectors
Modified CMP Analysis (Khoshnoudian and Kiani, 2012)
Generalized Pushover Analysis (Sucuoglu and Gunay, 2010)
Adaptive Pushover Procedure (Antoniou et al., 2002)
A Galaxy of Multi-mode Pushover Analysis
Procedures
Final Comprehensive Examination 19
The Uncoupled Modal Response History Analysis Procedure
20
-0.09
-0.07
-0.05
-0.03
-0.01
0.01
0.03
0.05
0.07
0.09
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
Cyclic Pushover
Monotonic Pushover in Positive Direction
Monotonic Pushover in Negative Direction
Brick walls begin to crack
Shear walls begin to crack
Columns
begin to
Crack
Shear wallβs steel reinforcement
bars begin to yield in tension
Shear wallβs steel bars begin
to yield in compression
Concrete crushing in
shear wall
Columnβs bars begin to
yield in compression
Columnβs steel bars
begin to yield in tension
Normalized Base
Shear (ππ1/π)
Roof Drift (π₯ππ/π»)
The Cyclic Behavior of
Case Study Buildings
44-story case study building in
Strong Direction
21
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.035 -0.015 0.005 0.025
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
19-story Building
Mode 133-story Building
Mode 1
44-story Building
Mode 1
Normalized Base Shear ( )
Roof Drift ( ) Roof Drift ( ) Roof Drift ( )
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.03 -0.01 0.01 0.03
-0.12
-0.08
-0.04
0
0.04
0.08
0.12
-0.025 -0.015 -0.005 0.005 0.015 0.025
-0.2
-0.1
0
0.1
0.2
-0.025 -0.015 -0.005 0.005 0.015 0.025
19-story Building
Mode 233-story Building
Mode 2
44-story Building
Mode 2
Normalized Base Shear ( )
Roof Drift ( ) Roof Drift ( ) Roof Drift ( )
19-story Building
Mode 333-story Building
Mode 3
44-story Building
Mode 3
Normalized Base Shear ( )
Roof Drift ( ) Roof Drift ( ) Roof Drift ( )
The Cyclic Behavior of
Case Study Buildings
20
20
20
Introducing AIT SolutionsFinal Comprehensive Examination 22
Idealization of Cyclic Pushover Curves
Energy dissipated by
hysteretic damping
Yield
Cracks
close
ππ2
Unloading
starts
π½πΉπ π¦π
Compression
yielding of steel
bars
1
π·π¦π
πΉπ π πΏπ
π·π
2
3
1
4
π·π’π
πΉπ π¦π/πΏπ
πΉπ π’π/πΏπ
Normalized Base Shear (ππ1/π)
Roof Drift (π₯ππ/π»)
20-story Building
Mode 1
Cyclic Pushover Curve
Response of an SDF system
under a ground motion
-0.05
-0.03
-0.01
0.01
0.03
0.05
-0.0075 -0.005 -0.0025 0 0.0025 0.005 0.0075
Introducing AIT SolutionsFinal Comprehensive Examination 23
Idealization of Cyclic Pushover Curves
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-0.015 -0.005 0.005 0.015
Mode 1 (B3)
Normalized Base Shear ( )
Roof Drift Ratio ( )
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
Normalized Base Shear ( )
Roof Drift Ratio ( )
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.02 -0.01 0 0.01 0.02
Normalized Base Shear ( )
Roof Drift Ratio ( )
Mode 1 (B2)Mode 1 (B1)
Cyclic pushover
curve
Idealized SDF
system
Ground Motion Set 4
Introducing AIT SolutionsFinal Comprehensive Examination 24
Idealization of Cyclic Pushover Curves
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.004 -0.002 0 0.002 0.004
Mode 2 (B3)
Roof Drift Ratio ( )
-0.12
-0.09
-0.06
-0.03
0
0.03
0.06
0.09
0.12
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
Roof Drift Ratio ( )
-0.08
-0.05
-0.02
0.01
0.04
0.07
-0.003 -0.002 -0.001 0 0.001 0.002 0.003
Roof Drift Ratio ( )
Mode 2 (B2)Mode 2 (B1)
Cyclic pushover
curve
Idealized SDF
system
Ground Motion Set 4
Introducing AIT SolutionsFinal Comprehensive Examination 25
Idealization of Cyclic Pushover Curves
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-0.006 -0.004 -0.002 0 0.002 0.004 0.006
Mode 3 (B3)
Roof Drift Ratio ( )
-0.12
-0.09
-0.06
-0.03
0
0.03
0.06
0.09
0.12
-0.003 -0.001 0.001 0.003
Roof Drift Ratio ( )
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.003 -0.001 0.001 0.003
Roof Drift Ratio ( )
Mode 3 (B2)Mode 3 (B1)
Cyclic pushover
curve
Idealized SDF
system
Ground Motion Set 4
26
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.005 -0.0025 0 0.0025 0.005
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.004 -0.002 0 0.002 0.004-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.006 -0.004 -0.002 0 0.002 0.004 0.006
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.002 -0.001 0 0.001 0.002
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-0.01 -0.005 0 0.005 0.01
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.002 -0.001 0 0.001 0.002
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-0.002 -0.001 0 0.001 0.002-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-0.002 -0.001 0 0.001 0.002
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
-0.01 -0.005 0 0.005 0.01
Mode 1 (B3) Mode 2 (B3) Mode 3 (B3)
Normalized Base Shear ( )
Roof Drift Ratio ( ) Roof Drift Ratio ( ) Roof Drift Ratio ( )
Normalized Base Shear ( )
Roof Drift Ratio ( ) Roof Drift Ratio ( ) Roof Drift Ratio ( )
Normalized Base Shear ( )
Roof Drift Ratio ( ) Roof Drift Ratio ( ) Roof Drift Ratio ( )
Mode 1 (B2) Mode 2 (B2) Mode 3 (B2)
Mode 1 (B1) Mode 2 (B1) Mode 3 (B1)
Cyclic pushover
curve
Idealized SDF
system
Cyclic pushover
curve
Idealized SDF
system
Cyclic pushover
curve
Idealized SDF
system
Ground Motion Set 1
The Idealization of Cyclic
Pushover Curves
27
Modal Decomposition of
Nonlinear Responses
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Peak Displacement (mm)
Mode 2
Mode 3
0
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8
Peak Inter-story Drift Ratio (%)
Mode 1
Mode 2
Mode 3
Envelope of
Combined History
Mode 1
Envelope of
Combined History Le
vel N
o.
44-story case study building in
Strong Direction
Ground Motion Set 4
28
0
5
10
15
20
25
30
35
40
45
0 15 30 45
Peak Story Shear (x 106 N)
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5
Peak Moment (x 106 KN m)
Mode 2
Mode 1
Mode 3
Envelope of
Combined History
Mode 2
Mode 1
Mode 3
Envelope of
Combined History
Level N
o.
Modal Decomposition of
Nonlinear Responses
44-story case study building in
Strong Direction
Ground Motion Set 4
Introducing AIT SolutionsFinal Comprehensive Examination 29
UMRHA vs. NLRHA
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50
Story Shear (x106 N)
Story Shear Envelope
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
Moment (x106 KN m)
Overturning Moment
0
5
10
15
20
25
30
35
40
45
0 200 400 600
No.
of S
tories
Displacement (mm)
Displacement Envelope
UMRHA (Combined3 Modes)
NLRHA
0
5
10
15
20
25
30
35
40
45
0 0.25 0.5 0.75 1
IDR (%)
Inter-story Drift Ratio
UMRHA(Combined 3 Modes)
NLRHA
44-story case study building in Strong Direction - Ground Motion Set 4
30
Modal Decomposition of
Nonlinear Responses
44-story case study building in
Strong Direction
Ground Motion Set 4
NLRHA
UMRHA (Combined)
Individual Modes
0
5
10
15
20
25
30
35
40
45
0 500 1000 1500 2000
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5
0
5
10
15
20
25
30
35
40
45
0 25 50 75 1000
5
10
15
20
25
30
35
40
45
0 1 2 3
0
5
10
15
20
25
30
35
0 1000 2000 3000
0
5
10
15
20
25
30
35
0 15 30 45 600
5
10
15
20
25
30
35
0 0.5 1 1.5 2
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3
0
5
10
15
20
0 500 1000 1500
0
5
10
15
20
0 10 20 30 40 500
5
10
15
20
0 0.2 0.4 0.6 0.8
0
5
10
15
20
0 0.5 1 1.5 2 2.5
B1
B2
B3
Displacement (mm) Inter-story Drift Ratio (%) Story Shear (x 106 N) Moment (x 106 KN m)
Le
ve
l N
o.
Le
ve
l N
o.
Le
ve
l N
o.
31
Modal Decomposition of
Nonlinear Responses
44-story case study building in
Strong Direction
Ground Motion Set 4
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
-600
-400
-200
0
200
400
600
800
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
-600
-400
-200
0
200
400
600
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
-150
-100
-50
0
50
100
150
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
-15
-10
-5
0
5
10
15
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
Ground Acceleration,
Time (sec)
Roof Displacement,
Time (sec)
Roof Displacement,
Time (sec)
Roof Displacement,
Time (sec)
Roof Displacement,
Time (sec)
Mode 1
Mode 2
Mode 3
CombinedUMRHA
NLRHA
Final Comprehensive Examination 32
An Extended Displacement Coefficient Method (EDCM)
(or A Simplified Modal Pushover Analysis Procedure)
Introducing AIT SolutionsFinal Comprehensive Examination 33
+π΄ππ
πππ
π₯ππ
+
π΄πππ΄ππ
+ β¦
Mu
lti-
mo
de
Pu
sh
ove
r
An
aly
sis
ππ = ππ,π2
πΏππ
ππ,π π₯π = πΏππ2
πΏππ = π€πβπΉπβπΈ
ππ΄,ππππ2
4π2π
Sim
pli
fie
d M
od
al
Pu
sh
ove
r
An
aly
sis
Pro
ce
du
re
π π¦,π =πΆπ,πππ΄,π
ππ¦,π π
π· π‘ = βπ΄π π₯π (π‘)
π₯π(π‘)
=
πππ
π₯ππ
ππππ¦
πΎππΌππΎπ
Actual
Idealized
βπΉπβπΈ
1
Mode 1Mode 2Mode 3
π
Pick βπΉπ/βπΈ ratios
for each mode using
its π π¦ and πΌ
Pushover curve and its
idealization, for each mode
Determine target
displacement (πΏππ) for each mode
Extract Peak response
for each modeCombine modal
responses
No
nlin
ea
r R
es
po
nse
His
tory
An
aly
sis
(N
LR
HA
)
Introducing AIT SolutionsFinal Comprehensive Examination 34
Determination of Displacement Coefficients
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
Mean of 6 GMs
(π π¦ = 2, ππ = 0.025)
π₯πΉππ₯πΈ
Time Period (sec)
Mean of 6 GMs
(π π¦ = 8, ππ = 0.025)
π₯πΉππ₯πΈ
Time Period (sec)
Mean of 6 GMs
(π π¦ = 2, ππ = 0.025)
π₯πΈπππ₯πΈ
Time Period (sec)
Mean of 6 GMs
(π π¦ = 8, ππ = 0.025)
π₯πΈπππ₯πΈ
Time Period (sec)
Set 1
Set 2
Set 3
35
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
Mean of 6 GMs
Set 2 ( )
Time Period (sec)
= 2
= 4
= 6= 8
Mean of 6 GMs
Set 1 ( )
Time Period (sec)
Mean of 6 GMs
Set 3 ( )
Time Period (sec)
Mean of 6 GMs
Set 2 ( )
Time Period (sec)
Mean of 6 GMs
Set 1 ( )
Time Period (sec)
Mean of 6 GMs
Set 3 ( )
Time Period (sec)
= 2
= 4
= 6= 8
= 2
= 4
= 6= 8
= 2
= 4
= 6= 8
= 2
= 4
= 6= 8
= 2
= 4
= 6= 8
Determination of
Displacement Coefficients
Effect of π π¦
Ground Motion Sets 1, 2 and 3
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
36
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
ππ€5.8 β 6.5, π ππ’π = 13 β 30 Km
ππ€5.8 β 6.5, π ππ’π = 30 β 60 Km
ππ€5.8 β 6.5, π ππ’π = 60 β 200 Km
ππ€6.6 β 6.9, π ππ’π = 13 β 30 Km
ππ€6.6 β 6.9, π ππ’π = 30 β 60 Km
ππ€6.6 β 6.9, π ππ’π = 60 β 200 Km
ππ€7.0 β 8.0, π ππ’π = 13 β 30 Km
ππ€7.0 β 8.0, π ππ’π = 30 β 60 Km
ππ€7.0 β 8.0, π ππ’π = 60 β 200 Km
NEHRP site class B
NEHRP site class C
NEHRP site class D
T (sec)
π π΄(n
orm
aliz
ed to
PG
A)
π π΄(n
orm
aliz
ed to
PG
A)
Additionally Selected
Typical Ground Motions
Determination of
Displacement Coefficients
Introducing AIT SolutionsFinal Comprehensive Examination 37
Determination of Displacement Coefficients
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
π₯πΈπππ₯πΈ
π₯πΉππ₯πΈ
ππ€7.0 β ππ€8.0, π ππ’π = 13 β 30 Km
Mean of 30 GMs (ππ = 0.05)
ππ€7.0 β ππ€8.0, π ππ’π = 13 β 30 Km
Mean of 30 GMs (ππ = 0.05)
Time Period (sec)Time Period (sec)
π π¦ = 2
π π¦ = 4
π π¦ = 6
π π¦ = 8
Introducing AIT SolutionsFinal Comprehensive Examination 38
Determination of Displacement Coefficients
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
ππ€7.0 β ππ€8.0, π ππ’π = 60 β 200 Km
Mean of 30 GMs
(π π¦ = 2, π½ = 0.2, π = 0.05)
ππ€5.8 β ππ€6.5, π ππ’π = 30 β 60 Km
Mean of 30 GMs
(π π¦ = 2, πΌ = 0.15, ππ = 0.05)
π₯πΉππ₯πΈ
π₯πΉππ₯πΈ
πΌ = 0πΌ = 0.15πΌ = 0.3πΌ = 0.5
Time Period (sec)
π½ = 0.2π½ = 0.4π½ = 0.6π½ = 0.8π½ = 1.0
Time Period (sec)
Introducing AIT SolutionsFinal Comprehensive Examination 39
Displacement Coefficients β Comparison with FEMA 440 (2005), ASCE/SEI 41-13 (2014)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 1 2 3 4
Mean of 30 GMs
(π π¦ = 2, ππ = 0.05)
π₯πΉππ₯πΈ
π₯πΈπππ₯πΈ
πΆ1 Site Class B
πΆ1 Site Class C
πΆ1 Site Class D
Mean of 30 GMs
(π π¦ = 2, ππ = 0.05)
πΆ1 Site Class B
πΆ1 Site Class C
πΆ1 Site Class D
FEMA 440 (2005),
ASCE/SEI 41-13 (2014)FEMA 440 (2005),
ASCE/SEI 41-13 (2014)
This Study This Study
Introducing AIT SolutionsFinal Comprehensive Examination 40
The EDCM β Individual Modal Demands β B3 β Ground Motion Sets 1, 2 and 3
0
5
10
15
20
25
30
35
40
45
0 10 20 30
Story Shear (x106 N)
Series3
Series4
0
5
10
15
20
25
30
35
40
45
0 10 20 30
Story Shear (x106 N)
Series3
Series4
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
Story Shear (x106 N)
Series3
Series4
Le
ve
l N
o.
UMRHA
EDCM
Ground Motion Set 1 Ground Motion Set 2 Ground Motion Set 3
UMRHA
EDCM
UMRHA
EDCM
Introducing AIT SolutionsFinal Comprehensive Examination 41
EDCM β Individual Modal Demands β Ground Motion Set 4
0
5
10
15
20
25
30
35
40
45
0 20 40 60
Story Shear (x106 N)
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20
Peak Story Shear (x106 N)
0
5
10
15
20
25
30
35
0 10 20 30 40
Peak Story Shear (x106 N)
EDCM
UMRHA
B1
EDCM
UMRHA
EDCM
UMRHA
B2 B3
Mode 3
Mode 1
Mode 2
Mode 3
Mode 1
Mode 2
Mode 3
Mode 1
Mode 2
Le
ve
l N
o.
Introducing AIT SolutionsFinal Comprehensive Examination 42
EDCM β Combined Demands β Ground Motion Set 4
0
2
4
6
8
10
12
14
16
18
20
0 10 20 30 40 500
2
4
6
8
10
12
14
16
18
20
0 300 600 900 1200
0
2
4
6
8
10
12
14
16
18
20
0 0.25 0.5 0.75 10
2
4
6
8
10
12
14
16
18
20
0 1 2 3
Peak Displacement (mm) Peak Inter-story Drift Ratio (%) Peak Story Shear (x 106 N) Peak Moment (x 106 KN m)
B1 NLRHA Mean
UMRHA Mean
EDCMLe
ve
l N
o.
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3
0
5
10
15
20
25
30
35
0 500 1000 1500 2000 2500
0
5
10
15
20
25
30
35
0 10 20 30 40 50 600
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5
0
5
10
15
20
25
30
35
40
45
0 25 50 75 1000
5
10
15
20
25
30
35
40
45
0 400 800 1200 1600 2000
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5 3
Peak Displacement (mm) Peak Inter-story Drift Ratio (%) Peak Story Shear (x 106 N) Peak Moment (x 106 KN m)
B2
B3
Le
ve
l N
o.
Le
ve
l N
o.
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
NLRHA Mean
UMRHA Mean
EDCM
44
Ground Motion Sets 1, 2 and 3
Performance of EDCM β
Combined Demands
B3
0
5
10
15
20
25
30
35
40
45
0 0.25 0.5 0.75
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 500
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
0
5
10
15
20
25
30
35
40
45
0 200 400 600 800
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
0
5
10
15
20
25
30
35
40
45
0 200 400 600 8000
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 300
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
0
5
10
15
20
25
30
35
40
45
0 0.1 0.2 0.3
NLRHA Mean
UMRHA Mean
EDCM
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5
Displacement (mm) Inter-story Drift Ratio (%) Story Shear (x 106 N) Moment (x 106 KN m)
45Final Comprehensive Examination
β’ The proposed EDCM works
β’ Requires significantly less computational time and effort compared to the detailed NLRHA procedure
β’ Can be considered as a convenient analysis option for the seismic performance evaluation of high-rise buildings
Summary
However
Final Comprehensive Examination 46
A Modified Response Spectrum Analysis (MRSA) Procedure
47
Linear Elastic Model
Determine Modal Properties
ππ, ππ, π€π
π1π2π3
ππ΄1
ππ΄2
ππ΄3
Sp
ectr
al A
cce
lera
tio
n (π π΄
)
Time Period (sec)
πππ =
π=1
π
π€π . ππ . ππ,π . ππ΄π
Determine Spectral Acceleration
for each Significant Mode
ππ1 ππ2 ππ3
πππ = (ππ1 )2 + (ππ2 )2 + (ππ3 )2 + β¦
Determine Equivalent Elastic Forces
Base S
hear
Roof Displacement
πππ
πππ
Maximum Displacement
during a design earthquake
ππ· = πππ =ππππ /πΌ
Determine Design Demands
π1 π2 π3
π = Response Modification Factor
For Initial
Viscous Dampingβ¦
N
Stories
Eigen-value Analysis
πβπ2π Ξ¦ = π
π₯π
For members not desired
to yield during a design
earthquake
Design Base Shear
ππ· = Ξ©ππππ
π₯πππ₯π =
πΆππ₯π
πΌ
πΌ = Importance Factor
Ξ© = Structural Over-strength Factor
πΆπ = Displacement Amplification Factor
The Standard RSA Procedure (ASCE 7-10, IBS 2012, EC 8)
Primary Motivation
48Final Comprehensive Examination
Criticisms on the RSA Procedure
Elastic Demands
Some FactorInelastic Demands =
Elastic Demands of That Mode
Same FactorInelastic Demands of Each Mode =
Statistical Combination of Modal Demands
(a)
(b)
(c)
Static Analysis Procedure (or Pseudo-dynamic, to say the least) (d)
Newmark & Hall (1982)
Eibl & Keintzel (1988)
Bertero (1986)
Miranda & Bertero (1994)
ATC 19 (1995)
ATC-34 (1995)
Cuesta & Aschheim (2001)
Priestley & Amaris (2002)
Foutch & Wilcoski (2005)
Sullivan, Priestley & Calvi (2008)
Maniatakis et al. (2013)
.
.
.
Introducing AIT SolutionsFinal Comprehensive Examination 49
βRay Clough and I regret we created the approximate response spectrum
method for seismic analysis of structures in 1962.β
ββ¦ allowed engineers to produce meaningless positive numbers of little or
no value.β
βDo not be called a Neanderthal man.β
CASE CLOSED
The use of the Response Spectrum Method in Earthquake Engineering
must be terminated.
It is not a dynamic analysis method β The results are not a function of time.
Criticisms on the RSA Procedure
β Edward L. Wilson
Professor Emeritus (CEE, UC Berkeley)
Introducing AIT SolutionsFinal Comprehensive Examination 50
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40 50 60
The Problem with R Factor
The elastic forces obtained from the
standard RSA procedure
The RSA elastic forces reduced by π
The inelastic forces obtained from the
NLRHA procedure
The actual reduction in RSA
elastic forces. The βrewardβ
of making a nonlinear model
The underestimation causing a βfalse
sense of safetyβ due to directly reducing
the RSA elastic forces by π factor
Story Shear (x106 N)
Sto
ry L
eve
l
51
Nonlinear Structure
πππ1, πππ1
β
+ + +β¦
F
DF
D
F
D
Mode 1Mode 2
Mode 3
+ + +β¦
Mode 1Mode 2
Mode 3
F
D
F
D
F
D
NLNL NL
ELEL
EL
πππ2, πππ2 πππ3, πππ3
β
NLRHA
UMRHA
MRSA
The Basic Concept of the
Modified Response
Spectrum Analysis (MRSA)
52
πΉπ π(π·π , π·π)/πΏπ
πππ,π2
ππ2
π·π,πππ₯
Energy Dissipation
by Hysteretic
Damping (πβ,π)
π·π
1
1
πππ,π = ΞΊππ + πβ,π
Total Energy Dissipation
by the Equivalent
Viscous Damping (πππ,π) πππ,π2
1
π·π,πππ₯ π·π
πΉπ π(π·π , π·π)/πΏπ
An Equivalent Linear System
with Elongated Period and
Additional Damping
A nonlinear SDF system with initial
circular natural frequency ππ, and with
initial inherent viscous damping ππ
Conversion of a Modal
Nonlinear SDF System in to
a Modal βEquivalent Linearβ
System
53
πβ,π =1
4π
)πΈπ·(π·π)πΈπ π(π·π
πΉπ π(π·π , π·π)/πΏπ
πΈπ π(π·π)
πΈπ·(π·π)
ππ ππ,π2
π·π,πππ₯
Energy Dissipation by
Hysteretic Damping (πβ,π)
π·π
1
Energy dissipated by hysteretic damping
in one force-deformation loop of an actual
nonlinear system
Energy dissipated by an equivalent
amount of viscous damping in one
harmonic cycle of an equivalent linear
system
=
Equal-Energy Assumption
Energy Dissipation by
Equivalent Viscous
Damping (ππΈπΏ,π)
πΉπ π,πππ₯/πΏπ
Determination of Hysteretic
Damping β Equal-energy
Dissipation Rule
54
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5
33-story (Mode 1)
Roof Drift ratio, π₯ππ/π» (%)
ππ ππ,π/ππ
20-story
(Mode 1)
44-story (Mode 1)
33 Story (Mode 2)
Secant Periods from Envelope of Cyclic Pushover Curves
44 Story (Mode 2)
20 Story (Mode 2)
33 Story (Mode 3)
44 Story
(Mode 3)
20 Story (Mode 3)
ππ ππ,π/ππ vs. Roof Drift Ratio (π₯ππ/π»)
βEquivalent Linearβ
Properties
55
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
πβ,π (%)
33-story
(Mode 1)
20-story
(Mode 1)
44-story
(Mode 1)
20-story (Mode 1) - Each
loop start from initial
unloaded condition
Hysteretic Damping from Area Enclosed by Cyclic Pushover Curves
Roof Drift ratio, π₯ππ/π» (%)
(b) πβ,π vs. Roof Drift Ratio (π₯ππ/π»)
βEquivalent Linearβ
Properties
56
π2
π 2
Spectr
al A
ccele
ratio
n (
SA
)
Time Period (sec)
π1
π 1
Spectr
al A
ccele
ratio
n (
SA
)Time Period (sec)
Linear Model
β + + + β¦
Mode 1Mode 2
Mode 3
F
D
F
D
F
D
π3
π 3
Spectr
al A
ccele
ratio
n (
SA
)
Time Period (sec)
π3,ππ
π 1,πππ 1,ππ
π 3,ππ
π1
π1,ππ
π2
π2,πππ3
π3,ππ
πππ/ππ πππ
π3,ππ
ππ
π2,ππ
ππ
π1,ππ
πππ1,ππ
π2,ππ
π3,ππ
π₯3π
Initial Viscous Damping
Initial Time Period
πππ/ππ vs. π₯ππ
and πππ vs. π₯ππ
Relationships
Mode 1
Mode 2Mode 3
Mode 1
Mode 2Mode 3
π1, π1, π€1, π1 π2, π2, π€2, π2 π3, π3, π€3, π3
Modal Analysis
π₯2π π₯1
π π₯ππ₯3π π₯2
π π₯1π π₯π
π₯π‘πππππ = π₯π,π πππ
π = ππ,π πππ . π€π . ππ·π
ππ·π =ππ2
4π2ππ΄π
π1,πππ1
π 1
π2,πππ2
π 1
π 3
π (sec) π (sec)
The Modified Response
Spectrum Analysis (MRSA)
What is βModifiedβ?
Introducing AIT SolutionsFinal Comprehensive Examination 57
The MRSA Procedure β Individual Modal Demands β Ground Motion Set 4
0
5
10
15
20
25
30
35
40
45
0 20 40 60 80
Peak Story Shear (x106 N)
Mode 3
Mode 1
Mode 2
0
5
10
15
20
25
30
35
0 10 20 30 40
Peak Story Shear (x106 N)
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20
Peak Story Shear (x106 N)
MRSA
UMRHA
Mode 3
Mode 1
Mode 2
Mode 3
Mode 1
Mode 2
Le
ve
l N
o.
B1
MRSA
UMRHA
MRSA
UMRHA
B2 B3
58
0
10
20
30
40
0 5 10 15 20
0
10
20
30
40
0 1 2 3 4
0
10
20
30
40
0 5 10 15 20 25
0
10
20
30
40
0 2 4 6 8 10
0
10
20
30
40
0 10 20 30
0
10
20
30
40
0 5 10 15 20
No
. o
f S
tories
0
10
20
30
40
0 5 10 15 20
0
10
20
30
40
0 10 20 30 40
0
10
20
30
40
0 5 10 15
No
. o
f S
tories
No
. o
f S
tories NLRHA
MRSA
Standard RSA
(R = 4.5)
Story Shear (x106 N)
Ground
Motion Set 1
Ground
Motion Set 2
Ground
Motion Set 3
Mode 1 Mode 2 Mode 3
Mode 1 Mode 2 Mode 3
Mode 1 Mode 2 Mode 3
Ground Motion Sets 1, 2 and 3
Performance of the MRSA
Procedure β Individual
Modal Demands
B3
Introducing AIT SolutionsFinal Comprehensive Examination 59
0
5
10
15
20
25
30
35
40
45
0 200 400 600 800
No
. o
f S
tori
es
Displacement (mm)
Displacement Envelope
0
5
10
15
20
25
30
35
40
45
0 0.25 0.5 0.75 1
IDR (%)
Inter-story Drift Ratio
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
Story Shear (x106 N)
Story Shear Envelope
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
Moment (x106 KN m)
Overturning Moment
Overall Performance of the MRSA Procedure β B3 β Ground Motion Set 1
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Introducing AIT SolutionsFinal Comprehensive Examination 60
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200
No
. o
f S
torie
s
Displacement (mm)
Displacement Envelope
0
5
10
15
20
25
30
35
40
45
0 0.05 0.1 0.15 0.2 0.25
IDR (%)
Inter-story Drift Ratio
0
5
10
15
20
25
30
35
40
45
0 0.25 0.5 0.75 1 1.25
Moment (x106 KN m)
Overturning Moment
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
Story Shear (x106 KN)
Story Shear Envelope
Overall Performance of the MRSA Procedure β B3 β Ground Motion Set 2
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Introducing AIT SolutionsFinal Comprehensive Examination 61
0
5
10
15
20
25
30
35
40
45
0 250 500 750
No
. o
f S
torie
s
Displacement (mm)
Displacement Envelope
0
5
10
15
20
25
30
35
40
45
0 0.15 0.3 0.45 0.6 0.75
IDR (%)
Inter-story Drift Ratio
0
5
10
15
20
25
30
35
40
45
0 10 20 30
Story Shear (x106 KN)
Story Shear Envelope
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
Moment (x106 KN m)
Overturning Moment
Overall Performance of the MRSA Procedure β B3 β Ground Motion Set 3
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
620
5
10
15
20
25
30
35
40
45
0 25 50 75 100
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2
0
5
10
15
20
25
30
35
40
45
0 500 1000 1500
0
5
10
15
20
25
30
35
0 1000 2000
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3
0
5
10
15
20
25
30
35
0 10 20 30 40 50 60
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3
0
5
10
15
20
0 10 20 30 40 50
0
5
10
15
20
0 0.5 1 1.5
0
5
10
15
20
0 300 600 900 1200
0
5
10
15
20
0 0.5 1 1.5 2 2.5 3
B1
B2
B3
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Standard RSA Γ Ξ©
Overall Performance of
MRSA
Ground Motion Set 4
Peak Displacement (mm) Peak Inter-story Drift Ratio (%) Peak Story Shear (x 106 N) Peak Moment (x 106 KN m)
63
0
5
10
15
20
25
30
35
40
45
-3000 -2000 -1000 0 1000 2000 3000
Moment (x106 N mm)
0
5
10
15
20
25
30
35
40
45
-3000 -2000 -1000 0 1000 2000 3000
Moment (x106 N mm)
No. of S
tories
MM
Ground Motion Set 4
B3
Bending Moments in
Columns
NLRHA Mean
MRSA
Standard RSA
Standard RSA Γ Ξ©
64
0
5
10
15
20
25
30
35
40
45
-50000 -25000 0 25000 50000
Moment (x106 N mm)
0
5
10
15
20
25
30
35
40
45
-20 -10 0 10 20
Shear (x106 N)
No
. o
f S
torie
s
V M
NLRHA Mean
MRSA
Standard RSA
Standard RSA Γ Ξ©Ground Motion Set 4
B3
Shear Forces and
Bending Moments in
Shear Walls
65Final Comprehensive Examination
β’ The proposed MRSA procedure works
β’ Requires significantly less computational time and effort compared to the detailed NLRHA procedure
β’ Simple, conceptually superior, provides mode-by-mode response, clear insight
β’ Doesnβt require nonlinear modeling
β’ Can be considered as a convenient analysis and design option
Summary and Conclusions
66Final Comprehensive Examination
β’ The UMRHA procedure
β’ The case study buildings
β’ The modal combination rule
β’ Development of generalized relationships for
β’ Displacement coefficients β The EDCM
β’ Equivalent linear properties β The MRSA procedure
Limitations and Further Study
Final Comprehensive Examination 67
Thank you for your attention