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
• Global-level Responses
• Inter-story Responses
• 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
… … … Simplified Estimation of Seismic Demands
Simplified Structural Models
Simplified Loading
Any combination of simplified approaches
Simplified Seismic Analysis
Procedures
Triangular
Modal Expansion
… … …
… … …
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
• Real case study buildings (20-, 33-,
and 44-story high)
• Different types and levels of input
ground motions
Conception and theoretical formulation
Application and subsequent validation
Practicality, limitations and sources of uncertainty
• Practical and convenient applicability
• Future improvements
• Modal Decomposition
• Higher-mode Effects
Introducing AIT SolutionsFinal Comprehensive Examination 6
Basic Tool to Achieve Study Objectives – The UMRHA Procedure
≅+ +…
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
≅
+ + …
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
• Located in Bangkok, Thailand
• Heights vary from 20 to 44 stories
• RC slab-column frames carry gravity loads
• RC walls & cores resist lateral loads
• Masonry infill walls extensively used
• Designed for wind loads, but not for seismic
effects
• 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
𝐸𝑜 𝐸𝑜 𝐸𝑜
Stress
Strain휀𝑐′ 휀𝑐𝑢
𝑓𝑐′
𝑓𝑡
휀𝑡
Mander Envelope
Perform 3D Envelope
Unloading
Reloading
𝐸𝑜
Stress
Strain휀𝑦 휀𝑢
𝑓𝑢
𝑓𝑦
휀𝑠ℎ
Park Envelope
Perform 3D
𝑑𝑐𝐸𝑜
𝛼𝐸𝑜
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 – Background
Introducing AIT SolutionsFinal Comprehensive Examination 12
Nonlinear Static Procedures
• Several EL Procedures (Individual studies)
• Capacity Spectrum Method (CSM) (ATC 40, FEMA 273, FEMA 356)
• FEMA 440 Improved EL Procedure
Equivalent Linearization
• Several expressions for displacement modification (Individual studies)
• Displacement Coefficient Method (ATC 40, FEMA 273, FEMA 356, FEMA 440, ASCE 41-06, ASCE 41-13)
Displacement Modification
Full 3D Nonlinear
MDF Model
𝐹𝑛
𝐹1
.
.
.
Monotonic Pushover
Analysis Procedure
𝑥𝑟
𝑉𝑏
Determination of
Pushover Curve
𝑉𝑏
𝑥𝑟
Idealization of
Pushover Curve
𝐹
𝐷
An Equivalent
SDF System
The Concept of an “Equivalent SDF System”
Introducing AIT SolutionsFinal Comprehensive Examination 13
Capacity Spectrum Method (ATC 40, 1996)
Full 3D Nonlinear MDF Model
• Determine the modal
properties 𝜙1, 𝑇, Γ and 𝐶𝑚
𝐹𝑛
𝐹1
.
.
.
Monotonic Pushover Analysis
• Monotonically push and get
the 𝑉𝑏1 vs. 𝑥1𝑟 relationship
𝑥1𝑟
𝑉𝑏1
𝑉𝑏1Determination of Responses
𝑥1𝑟
• Extract the responses from the pushover database at 𝛿𝑃
𝐹𝑛
𝐹1
.
.
.
𝛿𝑃
𝑉𝑏1𝛿𝑃
𝑉𝑏1
Conversion of Pushover Curve
to ADRS Format
𝑥1𝑟
𝑆𝐴 =𝑉𝑏 𝑊 𝐶𝑚
𝑆𝐴
𝑆𝐷
𝑆𝐷 =𝑇2
4𝜋2𝑆𝐴
𝑆𝐷 =𝑥𝑟
Γ𝜙1𝑟
Determination of Performance
Point𝑆𝐴
𝑆𝐷
Performance Point
Demand Spectrum
Capacity Spectrum
• The “Demand Spectrum” is the reduced
form of 5% damped spectrum
𝛿𝑃
Introducing AIT SolutionsFinal Comprehensive Examination 14
Displacement Coefficient Method (ATC 40, FEMA 273, FEMA 356)
𝛿𝑇 = 𝐶0𝐶1𝐶2𝐶3𝑆𝐴𝑇𝑒2
4𝜋2𝑔
“Effective” periodFor converting spectral
to roof displacement
For converting elastic to
inelastic displacement
To account for the shape
of hysteresis loopsSpectral acceleration at 𝑇𝑒
Target Displacement
To account for dynamic
𝑃 − Δ effects
Introducing AIT SolutionsFinal Comprehensive Examination 15
Improved NSPs – 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
𝑅𝑦 = 8
𝑅𝑦 = 6
𝑅𝑦 = 4
𝑅𝑦 = 3
𝑅𝑦 = 2
𝑅𝑦 = 1.5
𝐶1,𝐸𝑃𝑃 Site Class C – Mean of 20 Ground Motions
Two Basic Spectral Regions
𝐶1 is on average larger than 1
𝐶1 increases with decreasing 𝑇𝐶1 increases with increasing 𝑅𝑦
𝐶1 is approximately constant with changes in 𝑇𝐶1 doesn’t change much with changes in 𝑅𝑦𝐶1 is on average approximately equal to 1
𝑇 (sec)
Introducing AIT SolutionsFinal Comprehensive Examination 17
ASCE/SEI 41-13 (2013) Nonlinear Static Procedure
Full 3D Nonlinear MDF Model
• Determine the modal
properties 𝜙1, 𝑇, Γ and 𝐶𝑚
𝐹𝑛
𝐹1
.
.
.
Monotonic Pushover Analysis
• Monotonically push and get
the 𝑉𝑏1 vs. 𝑥1𝑟 relationship
𝑥1𝑟
𝑉𝑏1
𝑉𝑏1
Idealization of Pushover Curve
𝑥1𝑟
IdealizedActual
• Idealized force-displacement
relationship.
• Determine 𝐾𝑒, 𝑉𝑦 and 𝑅𝑦
Determination of Target
Displacement
• Determine the effective time
period and coefficients
• Target displacement
𝛿𝑇 = 𝐶𝑜𝐶1𝐶2𝑆𝐴𝑇𝑒2
4𝜋2𝑔
𝑇𝑒 = 𝑇𝑖𝐾𝑒𝐾𝑖
𝐶1 = 1 +𝑅𝑦 − 1
𝑎𝑇𝑒2
𝐶2 = 1 +1
800
𝑅𝑦 − 1
𝑇𝑒
2
𝑉𝑏1Determination of Responses
𝑥1𝑟
• Extract the responses from the pushover database at 𝛿𝑇
𝐹𝑛
𝐹1
.
.
.
𝛿𝑇
𝑉𝑏1𝛿𝑇
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 wall’s steel reinforcement
bars begin to yield in tension
Shear wall’s steel bars begin
to yield in compression
Concrete crushing in
shear wall
Column’s bars begin to
yield in compression
Column’s steel bars
begin to yield in tension
Normalized Base
Shear (𝑉𝑏1/𝑊)
Roof Drift (𝑥𝑖𝑟/𝐻)
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
𝜔𝑖2
Unloading
starts
𝛽𝐹𝑠𝑦𝑖
Compression
yielding of steel
bars
1
𝐷𝑦𝑖
𝐹𝑠𝑖 𝐿𝑖
𝐷𝑖
2
3
1
4
𝐷𝑢𝑖
𝐹𝑠𝑦𝑖/𝐿𝑖
𝐹𝑠𝑢𝑖/𝐿𝑖
Normalized Base Shear (𝑉𝑏1/𝑊)
Roof Drift (𝑥𝑖𝑟/𝐻)
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
+𝑴𝝓𝟏
𝑉𝑏𝑖
𝑥𝑖𝑟
+
𝑴𝝓𝟐𝑴𝝓𝟑
+ …
Mu
lti-
mo
de
Pu
sh
ove
r
An
aly
sis
𝑟𝑉 = 𝑟𝑉,𝑖2
𝛿𝑇𝑖
𝑟𝑉,𝑖 𝑥𝑟 = 𝛿𝑇𝑖2
𝛿𝑇𝑖 = 𝛤𝑖∆𝐹𝑆∆𝐸
𝑆𝐴,𝑖𝑇𝑒𝑖2
4𝜋2𝑔
Sim
pli
fie
d M
od
al
Pu
sh
ove
r
An
aly
sis
Pro
ce
du
re
𝑅𝑦,𝑖 =𝐶𝑚,𝑖𝑆𝐴,𝑖
𝑉𝑦,𝑖 𝑊
𝑷 𝑡 = −𝑴𝒍 𝑥𝑔 (𝑡)
𝑥𝑔(𝑡)
=
𝑉𝑏𝑖
𝑥𝑖𝑟
𝑉𝑏𝑖𝑦
𝐾𝑖𝛼𝑖𝐾𝑖
Actual
Idealized
∆𝐹𝑆∆𝐸
1
Mode 1Mode 2Mode 3
𝑇
Pick ∆𝐹𝑆/∆𝐸 ratios
for each mode using
its 𝑅𝑦 and 𝛼
Pushover curve and its
idealization, for each mode
Determine target
displacement (𝛿𝑇𝑖) 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
(𝑅𝑦 = 2, 𝜉𝑖 = 0.025)
𝑥𝐹𝑆𝑥𝐸
Time Period (sec)
Mean of 6 GMs
(𝑅𝑦 = 8, 𝜉𝑖 = 0.025)
𝑥𝐹𝑆𝑥𝐸
Time Period (sec)
Mean of 6 GMs
(𝑅𝑦 = 2, 𝜉𝑖 = 0.025)
𝑥𝐸𝑃𝑃𝑥𝐸
Time Period (sec)
Mean of 6 GMs
(𝑅𝑦 = 8, 𝜉𝑖 = 0.025)
𝑥𝐸𝑃𝑃𝑥𝐸
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 𝑅𝑦
Ground Motion Sets 1, 2 and 3
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 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
𝑀𝑤5.8 – 6.5, 𝑅𝑟𝑢𝑝 = 13 – 30 Km
𝑀𝑤5.8 – 6.5, 𝑅𝑟𝑢𝑝 = 30 – 60 Km
𝑀𝑤5.8 – 6.5, 𝑅𝑟𝑢𝑝 = 60 – 200 Km
𝑀𝑤6.6 – 6.9, 𝑅𝑟𝑢𝑝 = 13 – 30 Km
𝑀𝑤6.6 – 6.9, 𝑅𝑟𝑢𝑝 = 30 – 60 Km
𝑀𝑤6.6 – 6.9, 𝑅𝑟𝑢𝑝 = 60 – 200 Km
𝑀𝑤7.0 – 8.0, 𝑅𝑟𝑢𝑝 = 13 – 30 Km
𝑀𝑤7.0 – 8.0, 𝑅𝑟𝑢𝑝 = 30 – 60 Km
𝑀𝑤7.0 – 8.0, 𝑅𝑟𝑢𝑝 = 60 – 200 Km
NEHRP site class B
NEHRP site class C
NEHRP site class D
T (sec)
𝑆 𝐴(n
orm
aliz
ed to
PG
A)
𝑆 𝐴(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
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 8
𝑥𝐸𝑃𝑃𝑥𝐸
𝑥𝐹𝑆𝑥𝐸
𝑀𝑤7.0 – 𝑀𝑤8.0, 𝑅𝑟𝑢𝑝 = 13 – 30 Km
Mean of 30 GMs (𝜉𝑖 = 0.05)
𝑀𝑤7.0 – 𝑀𝑤8.0, 𝑅𝑟𝑢𝑝 = 13 – 30 Km
Mean of 30 GMs (𝜉𝑖 = 0.05)
Time Period (sec)Time Period (sec)
𝑅𝑦 = 2
𝑅𝑦 = 4
𝑅𝑦 = 6
𝑅𝑦 = 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
𝑀𝑤7.0 – 𝑀𝑤8.0, 𝑅𝑟𝑢𝑝 = 60 – 200 Km
Mean of 30 GMs
(𝑅𝑦 = 2, 𝛽 = 0.2, 𝜉 = 0.05)
𝑀𝑤5.8 – 𝑀𝑤6.5, 𝑅𝑟𝑢𝑝 = 30 – 60 Km
Mean of 30 GMs
(𝑅𝑦 = 2, 𝛼 = 0.15, 𝜉𝑖 = 0.05)
𝑥𝐹𝑆𝑥𝐸
𝑥𝐹𝑆𝑥𝐸
𝛼 = 0𝛼 = 0.15𝛼 = 0.3𝛼 = 0.5
Time Period (sec)
𝛽 = 0.2𝛽 = 0.4𝛽 = 0.6𝛽 = 0.8𝛽 = 1.0
Time Period (sec)
Introducing AIT SolutionsFinal Comprehensive Examination 39
Displacement Coefficients – 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
(𝑅𝑦 = 2, 𝜉𝑖 = 0.05)
𝑥𝐹𝑆𝑥𝐸
𝑥𝐸𝑃𝑃𝑥𝐸
𝐶1 Site Class B
𝐶1 Site Class C
𝐶1 Site Class D
Mean of 30 GMs
(𝑅𝑦 = 2, 𝜉𝑖 = 0.05)
𝐶1 Site Class B
𝐶1 Site Class C
𝐶1 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 – Individual Modal Demands – B3 – 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 – Individual Modal Demands – 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 – Combined Demands – 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 –
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
• The proposed EDCM works
• Requires significantly less computational time and effort compared to the detailed NLRHA procedure
• 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
𝑇𝑖, 𝜙𝑖, 𝛤𝑖
𝑇1𝑇2𝑇3
𝑆𝐴1
𝑆𝐴2
𝑆𝐴3
Sp
ectr
al A
cce
lera
tio
n (𝑆 𝐴
)
Time Period (sec)
𝑉𝑏𝑖 =
𝑛=1
𝑁
𝛤𝑖 . 𝑚𝑛 . 𝜙𝑖,𝑛 . 𝑆𝐴𝑖
Determine Spectral Acceleration
for each Significant Mode
𝑉𝑏1 𝑉𝑏2 𝑉𝑏3
𝑉𝑒𝑙 = (𝑉𝑏1 )2 + (𝑉𝑏2 )2 + (𝑉𝑏3 )2 + …
Determine Equivalent Elastic Forces
Base S
hear
Roof Displacement
𝑉𝑒𝑙
𝑉𝑖𝑛
Maximum Displacement
during a design earthquake
𝑉𝐷 = 𝑉𝑖𝑛 =𝑉𝑒𝑙𝑅/𝐼
Determine Design Demands
𝜙1 𝜙2 𝜙3
𝑅 = Response Modification Factor
For Initial
Viscous Damping…
N
Stories
Eigen-value Analysis
𝕂−𝜔2𝕄 Φ = 𝕆
𝑥𝑟
For members not desired
to yield during a design
earthquake
Design Base Shear
𝑉𝐷 = Ω𝑉𝑒𝑙𝑅
𝑥𝑚𝑎𝑥𝑟 =
𝐶𝑑𝑥𝑟
𝐼
𝐼 = Importance Factor
Ω = Structural Over-strength Factor
𝐶𝑑 = 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
“Ray Clough and I regret we created the approximate response spectrum
method for seismic analysis of structures in 1962.”
“… allowed engineers to produce meaningless positive numbers of little or
no value.”
“Do not be called a Neanderthal man.”
CASE CLOSED
The use of the Response Spectrum Method in Earthquake Engineering
must be terminated.
It is not a dynamic analysis method – The results are not a function of time.
Criticisms on the RSA Procedure
─ 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 𝑅
The inelastic forces obtained from the
NLRHA procedure
The actual reduction in RSA
elastic forces. The “reward”
of making a nonlinear model
The underestimation causing a “false
sense of safety” due to directly reducing
the RSA elastic forces by 𝑅 factor
Story Shear (x106 N)
Sto
ry L
eve
l
51
Nonlinear Structure
𝑇𝑒𝑞1, 𝜉𝑒𝑞1
≅
+ + +…
F
DF
D
F
D
Mode 1Mode 2
Mode 3
+ + +…
Mode 1Mode 2
Mode 3
F
D
F
D
F
D
NLNL NL
ELEL
EL
𝑇𝑒𝑞2, 𝜉𝑒𝑞2 𝑇𝑒𝑞3, 𝜉𝑒𝑞3
≅
NLRHA
UMRHA
MRSA
The Basic Concept of the
Modified Response
Spectrum Analysis (MRSA)
52
𝐹𝑠𝑖(𝐷𝑖 , 𝐷𝑖)/𝐿𝑖
𝜔𝑒𝑞,𝑖2
𝜔𝑖2
𝐷𝑖,𝑚𝑎𝑥
Energy Dissipation
by Hysteretic
Damping (𝜉ℎ,𝑖)
𝐷𝑖
1
1
𝜉𝑒𝑞,𝑖 = κ𝜉𝑖 + 𝜉ℎ,𝑖
Total Energy Dissipation
by the Equivalent
Viscous Damping (𝜉𝑒𝑞,𝑖) 𝜔𝑒𝑞,𝑖2
1
𝐷𝑖,𝑚𝑎𝑥 𝐷𝑖
𝐹𝑠𝑖(𝐷𝑖 , 𝐷𝑖)/𝐿𝑖
An Equivalent Linear System
with Elongated Period and
Additional Damping
A nonlinear SDF system with initial
circular natural frequency 𝜔𝑖, and with
initial inherent viscous damping 𝜉𝑖
Conversion of a Modal
Nonlinear SDF System in to
a Modal “Equivalent Linear”
System
53
𝜉ℎ,𝑖 =1
4𝜋
)𝐸𝐷(𝐷𝑖)𝐸𝑠𝑜(𝐷𝑖
𝐹𝑠𝑖(𝐷𝑖 , 𝐷𝑖)/𝐿𝑖
𝐸𝑠𝑜(𝐷𝑖)
𝐸𝐷(𝐷𝑖)
𝜔𝑠𝑒𝑐,𝑖2
𝐷𝑖,𝑚𝑎𝑥
Energy Dissipation by
Hysteretic Damping (𝜉ℎ,𝑖)
𝐷𝑖
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 (𝜉𝐸𝐿,𝑖)
𝐹𝑠𝑖,𝑚𝑎𝑥/𝐿𝑖
Determination of Hysteretic
Damping – 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, 𝑥𝑖𝑟/𝐻 (%)
𝑇𝑠𝑒𝑐,𝑖/𝑇𝑖
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)
𝑇𝑠𝑒𝑐,𝑖/𝑇𝑖 vs. Roof Drift Ratio (𝑥𝑖𝑟/𝐻)
“Equivalent Linear”
Properties
55
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
𝜉ℎ,𝑖 (%)
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, 𝑥𝑖𝑟/𝐻 (%)
(b) 𝜉ℎ,𝑖 vs. Roof Drift Ratio (𝑥𝑖𝑟/𝐻)
“Equivalent Linear”
Properties
56
𝑇2
𝑆 2
Spectr
al A
ccele
ratio
n (
SA
)
Time Period (sec)
𝑇1
𝑆 1
Spectr
al A
ccele
ratio
n (
SA
)Time Period (sec)
Linear Model
≅ + + + …
Mode 1Mode 2
Mode 3
F
D
F
D
F
D
𝑇3
𝑆 3
Spectr
al A
ccele
ratio
n (
SA
)
Time Period (sec)
𝑇3,𝑒𝑞
𝑆 1,𝑒𝑞𝑆 1,𝑒𝑞
𝑆 3,𝑒𝑞
𝜉1
𝜉1,𝑒𝑞
𝜉2
𝜉2,𝑒𝑞𝜉3
𝜉3,𝑒𝑞
𝑇𝑒𝑞/𝑇𝑖 𝜉𝑒𝑞
𝑇3,𝑒𝑞
𝑇𝑖
𝑇2,𝑒𝑞
𝑇𝑖
𝑇1,𝑒𝑞
𝑇𝑖𝜉1,𝑒𝑞
𝜉2,𝑒𝑞
𝜉3,𝑒𝑞
𝑥3𝑟
Initial Viscous Damping
Initial Time Period
𝑇𝑒𝑞/𝑇𝑖 vs. 𝑥𝑖𝑟
and 𝜉𝑒𝑞 vs. 𝑥𝑖𝑟
Relationships
Mode 1
Mode 2Mode 3
Mode 1
Mode 2Mode 3
𝑇1, 𝜙1, 𝛤1, 𝑚1 𝑇2, 𝜙2, 𝛤2, 𝑚2 𝑇3, 𝜙3, 𝛤3, 𝑚3
Modal Analysis
𝑥2𝑟 𝑥1
𝑟 𝑥𝑟𝑥3𝑟 𝑥2
𝑟 𝑥1𝑟 𝑥𝑟
𝑥𝑡𝑟𝑖𝑎𝑙𝑟 = 𝑥𝑖,𝑅𝑜𝑜𝑓
𝑟 = 𝜙𝑖,𝑅𝑜𝑜𝑓 . 𝛤𝑖 . 𝑆𝐷𝑖
𝑆𝐷𝑖 =𝑇𝑖2
4𝜋2𝑆𝐴𝑖
𝑇1,𝑒𝑞𝑇1
𝑆 1
𝑇2,𝑒𝑞𝑇2
𝑆 1
𝑆 3
𝑇 (sec) 𝑇 (sec)
The Modified Response
Spectrum Analysis (MRSA)
What is “Modified”?
Introducing AIT SolutionsFinal Comprehensive Examination 57
The MRSA Procedure – Individual Modal Demands – 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 – 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 – B3 – 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 – B3 – 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 – B3 – 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 × Ω
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 × Ω
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 × ΩGround Motion Set 4
B3
Shear Forces and
Bending Moments in
Shear Walls
65Final Comprehensive Examination
• The proposed MRSA procedure works
• Requires significantly less computational time and effort compared to the detailed NLRHA procedure
• Simple, conceptually superior, provides mode-by-mode response, clear insight
• Doesn’t require nonlinear modeling
• Can be considered as a convenient analysis and design option
Summary and Conclusions
66Final Comprehensive Examination
• The UMRHA procedure
• The case study buildings
• The modal combination rule
• Development of generalized relationships for
• Displacement coefficients – The EDCM
• Equivalent linear properties – The MRSA procedure
Limitations and Further Study
Final Comprehensive Examination 67
Thank you for your attention