Fawad A. NajamAsian Institute of Technology (AIT)
Thailand
A Modified Response Spectrum Analysis Procedure (MRSA) to Determine Nonlinear Demands of Tall Buildings
2Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Vibration Mode Shapes
3Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
From where it all started …
Lord Rayleigh (John William Strutt, 1842 - 1919)Joseph-Louis Lagrange (1736 - 1830)D’Alembert (1717–1783)
4Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Entrance in Earthquake Engineering
Arturo Danusso (1880 - 1968) Maurice Anthony Biot (1905 - 1985)Theodore von Karman (1881 - 1963)
5Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
The Standard RSA Procedure
Raw W. Clough (1920 - 2016) Edward L. Wilson (Born 1931)Edward L. Wilson (Born 1931)
On the Standardization …
As the size and complexity of projects increased, … it became
desirable and even necessary to … set up a series of routine
procedures for analysis and design.
– Hardy Cross
Engineers and Ivory Towers, 1952
With these standardized formulas and specifications and methods it
became possible to use a greater number of men and men with less
training to produce engineering works …
Standardization … as a check on fools and rascals or set up as an
intellectual assembly line, has served well in the engineering world”
8Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Linear Elastic Model
Determine Modal Properties
𝑇𝑖, 𝜙𝑖, 𝛤𝑖
𝑇1𝑇2𝑇3
𝑆𝐴1
𝑆𝐴2
𝑆𝐴3
Sp
ectr
al A
cce
lera
tio
n (
SA
)
Time Period (sec)
𝑉𝑏𝑛 =
𝑖=1
𝑁
𝛤𝑛 . 𝑚𝑖 . 𝜙𝑖,𝑛 . 𝑆𝐴𝑛
Determine Spectral Acceleration
for each Significant Mode
𝑉𝑏1 𝑉𝑏2 𝑉𝑏3
𝑉𝑒𝑙 = (𝑉𝑏1 )2 + (𝑉𝑏2 )2 + (𝑉𝑏3 )2 + …
Determine Elastic Base Shears
Forc
e
Displacement
𝐹𝑒𝑙
𝐹𝑖𝑛
∆𝑒𝑙= ∆𝑖𝑛
𝑉𝑖𝑛 =𝑉𝑒𝑙𝑅
Reduce Elastic Base Shear to account for
inelasticity
𝜙1 𝜙2 𝜙3
𝑅 = Response Modification
Factor, ASCE 7 (or
Behavior Factor, EC 8)
For Initial
Viscous Damping…
N
Stories
Eigen-value Analysis
𝕂− 𝜔2𝕄 Φ = 𝕆
The Standard RSA Procedure (ASCE 7-10, IBS 2012, EC 8)
9Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Solutions
Reducing each mode with same factor is wrong
“Each Mode undergo different level of nonlinearity”
Modified Modal Superposition(Priestley and Amaris, 2002)
R applied to only first mode, with higher modes assumed elastic
Shear Amplification Factors(EC 8) and Over-strength
Factors (ASCE 7-10)
(Sullivan, Priestley and Calvi, 2002)Higher-Mode properties with little (or
no) base rotational stiffness
(Maniatakis et al., 2013)Application of a different R for
each vibration mode
10Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Is it justified to “equally” modify the response of each vibration mode?
Lets separate them and check one-by-one
R ???
Our Approach
11Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
The Equivalent Single-degree-of-freedom (SDF) System
m, ξi
12Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
The Uncoupled Modal Response History Analysis
≅+ + + + + + …
14Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Each Lollipop is Different …
≅+ + + + + + …
15Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Each Lollipop is Different …
-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
0.2
-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
Cyclic modal pushover response of MDF System
Response of idealized SDF system
Mode 1
(Strong Direction)
Mode 2
(Strong Direction)
Mode 3
(Strong Direction)𝑉𝑏
Roof Drift (∆ )
𝑉𝑏 𝑉𝑏
Roof Drift (∆ ) Roof Drift (∆ )
16Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
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
17Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Modal Decomposition of Nonlinear Response
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Level N
o.
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
44-story case study building in Strong Direction
18Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Modal Decomposition of Nonlinear Response
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
44-story case study building in Strong Direction
The Basic Concept of Modified Response Spectrum Analysis (MRSA)
Nonlinear Structure
Detailed 3D Nonlinear Response History Analysis (NLRHA)
𝑇𝑒𝑞1, 𝜉𝑒𝑞1
≅
+ + +…
F
DF
D
F
D
Mode 1Mode 2
Mode 3
Uncoupled Modal Response History Analysis (UMRHA)
+ + +…
Mode 1Mode 2
Mode 3
Modified Response Spectrum Analysis (MRSA)
F
D
F
D
F
D
NLNL NL
ELEL
EL
𝑇𝑒𝑞2, 𝜉𝑒𝑞2 𝑇𝑒𝑞3, 𝜉𝑒𝑞3
≅
NLRHA
UMRHA
MRSA
20Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Solutions
Reducing each mode with same factor is wrong
“Each Mode undergo different level of nonlinearity”
Modified Modal Superposition(Priestley and Amaris, 2002)
R applied to only first mode, with higher modes assumed elastic
Shear Amplification Factors(EC 8) and Over-strength
Factors (ASCE 7-10)
(Sullivan, Priestley and Calvi, 2002)Higher-Mode properties with little (or
no) base rotational stiffness
(Maniatakis et al., 2013)Application of a different R for
each vibration mode
Equivalent Linearization
Why Modified Response Spectrum Analysis (MRSA)?
22Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
≅
𝑥𝑜
K𝑒𝑞
Total 𝜉𝑒𝑞 = Sum of Equivalent
Inherent Damping and
Equivalent Hysteretic Damping
Equivalent Viscous
Damping (𝜉𝑒𝑞)
An Equivalent Linear System with Elongated Period and Additional Damping
Force
𝐸𝑠𝑜(𝑥)
𝐸𝐷(𝑥)
𝜉ℎ =1
4𝜋
)𝐸𝐷(𝑥
)𝐸𝑠𝑜(𝑥=
)𝐸𝐷(𝑥
2𝜋 K𝑒𝑥𝑜2
K𝑒𝑞
Energy dissipated by total
damping in a Nonlinear
System
Energy dissipated by
equivalent viscous damping in
an equivalent linear system
=
K𝑖
𝑥𝑜
Hysteretic
Damping (𝜉ℎ)
A Nonlinear SystemConverted to
Equal-Energy Assumption
Displacement
Force
Displacement
Conversion of a Nonlinear System in to an “Equivalent Linear” System
23Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Natural Period Elongation – Monotonic Pushover Analysis
3D View
A 44-story Case Study Building
Elevation View
Pushover Curve (Strong Direction)
Base Shear vs. Roof Drift Ratio
No Crack
Cracked
24Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Hysteretic Damping – Cyclic Pushover Analysis – Strong Direction
-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 0.03-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.025 -0.015 -0.005 0.005 0.015 0.025
-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 (𝑉𝑏 ) 𝑉𝑏 𝑉𝑏
Roof Drift (∆ ) Roof Drift (∆ ) Roof Drift (∆ )
20-story 33-story 44-story
25Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
2.5
5
7.5
10
12.5
15
17.5
20
0 0.5 1 1.5 2 2.5
Equivalent Viscous Damping for 44-story Case Study Building in Strong Direction, from Cyclic Pushover
Analysis
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5
Equivalent Linear Periods of 44-story Case Study Building in Strong Direction, from Monotonic Pushover
Analysis
Roof Drift (%)
Mode 3Mode 2
Mode 1
𝑇𝑒𝑞 𝑇
Mode 1
Mode 3 Mode 2
𝜉𝑒𝑞(%)
Roof Drift (%)
𝜉𝑒𝑞 = 𝜉𝑖 + 𝜉ℎ
“Equivalent Linear” Properties – 44-story Case Study Building
26Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
5
10
15
20
25
30
0 0.5 1 1.5 2 2.5 3
Mode 1 Equivalent Viscous Damping from Cyclic Pushover Analysis
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2 2.5
Mode 1 Secant Period from Monotonic Pushover Analysis
𝑇𝑒𝑞 𝑇𝑖
44-story
20-story
33-story
𝜉ℎ(%)
20-story
33-story
44-story
Roof Drift (%) Roof Drift (%)
“Equivalent Linear” Properties
Equivalent Time Period Equivalent Damping
27Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Ground Motions
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5 2 2.5 3 3.5 4
Spectr
al A
ccele
ration
(g)
Time Period (sec)
Short-Period Target Spectrum and
Matched Ground Motions – Set 2
UHS Target Spectrum and Matched
Ground Motions – Set 1
Long-Period Target Spectrum
and Matched Ground Motions
– Set 3
Target
Mean of Matched
Individual Ground Motions
28Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20
Story Shear (x106 N)
Mode 1
UMRHA
MRSA
Standard RSA (R = 4.5)
0
5
10
15
20
25
30
35
40
45
0 10 20 30 40
Story Shear (x106 N)
Mode 2
0
5
10
15
20
25
30
35
40
45
0 5 10 15
Story Shear (x106 N)
Mode 3
No
. o
f S
torie
s
Modal Story Shears of a 44-story case study building in Strong Direction
Performance of MRSA at Individual Mode Level – B04 under EQ 1
29Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
5
10
15
20
25
30
35
40
45
0 2 4 6 8 10
Story Shear (x106 N)
Mode 1
UMRHA
MRSA
Standard RSA (R = 4.5)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30
Story Shear (x106 N)
Mode 2
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20
Story Shear (x106 N)
Mode 3
No
. o
f S
torie
sPerformance of MRSA at Individual Mode Level – B04 under EQ 2
Modal Story Shears of a 44-story case study building in Strong Direction
30Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20
Story Shear (x106 N)
Mode 2
0
5
10
15
20
25
30
35
40
45
0 1 2 3 4
Story Shear (x106 N)
Mode 3
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25
Story Shear (x106 N)
Mode 1
UMRHA
MRSA
Standard RSA (R = 4.5)
No
. o
f S
torie
sPerformance of MRSA at Individual Mode Level – B04 under EQ 3
Modal Story Shears of a 44-story case study building in Strong Direction
31Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
0
5
10
15
20
25
30
35
40
45
0 200 400 600 800
No
. o
f S
torie
s
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
44-story case study building in Strong Direction
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Overall Performance of MRSA – EQ Set 1
32Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
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
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Overall Performance of MRSA – EQ Set 2
44-story case study building in Strong Direction
33Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
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
NLRHA
MRSA
Standard RSA
(Cd = 4, R = 4.5)
Overall Performance of MRSA – EQ Set 3
44-story case study building in Strong Direction
34Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Local Response - Bending Moment in Columns (EQ Set 1)
0
5
10
15
20
25
30
35
40
45
-750 -500 -250 0 250 500 750
Moment (x106 N mm)
0
5
10
15
20
25
30
35
40
45
-750 -500 -250 0 250 500 750
Moment (x106 N mm)
Individual Ground Motions
NLRHA Mean
MRSA
Standard RSA (R = 4.5)
Standard RSA × Ω
No
. o
f S
torie
s
V
MM
44-story case study building (Strong Direction)
35Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Local Response – Shear Walls (EQ Set 1)
0
5
10
15
20
25
30
35
40
45
-15000-10000 -5000 0 5000 10000 15000
Moment (x106 N mm)
0
5
10
15
20
25
30
35
40
45
-1.5 -1 -0.5 0 0.5 1 1.5
Shear (x106 N)
No
. o
f S
torie
s V M Individual Ground Motions
NLRHA Mean
MRSA
Standard RSA (R = 4.5)
Standard RSA × Ω
44-story case study building (Strong Direction)
36Design of Tall Buildings: Trends and Advancements for Structural Performance (7 – 12 November 2016, Bangkok)
Positives of MRSA
• Linear Static Procedure
• Modal Combination Rule (Still carry the “bad name” of RSA)
• Accuracy of Local Response
Positives of MRSA
• Conceptually Superior, Expected to provide reasonably accurate estimates of nonlinear demands
• Simple, Provides mode-by-mode Response, Clear Insight
• Doesn’t require nonlinear modeling
Summary of MRSA
Elastic Model
Determine Modal
Properties
𝑇𝑖, 𝜙𝑖, 𝛤𝑖
𝑇1𝑇2𝑇3
𝑆𝐷1𝑆𝐷2
𝑆𝐷3
Sp
ectr
al D
isp
lacem
ent
(SD
)
Time Period (sec)
𝑉𝑒𝑞 𝑖 = 𝛤𝑖 . 𝑚𝑖 . 𝜙𝑖 . 𝑆𝐴(𝑇𝑒𝑞 𝑖 , 𝜉𝑒𝑞 𝑖)
Determine Spectral
Displacement for each
Significant Mode
𝑉𝑒𝑞1 𝑉𝑒𝑞2 𝑉𝑒𝑞3 𝑉𝑒𝑞4
𝑉𝑒𝑞 = (𝑉𝑒𝑞 1 )2 + (𝑉𝑒𝑞 2 )2 + (𝑉𝑒𝑞 3 )2 + …
Determine Equivalent Linear Response for each Mode
𝜙1 𝜙2 𝜙3 𝜙4
Amplitude
𝑇𝑒𝑞 𝜉𝑒𝑞
Δ𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = Δ𝑖,𝑅𝑜𝑜𝑓 = 𝜙𝑖,𝑅𝑜𝑜𝑓 . 𝛤𝑖 . 𝑆𝐷𝑖
𝑇𝑒𝑞3
𝑇𝑒𝑞2
𝑇𝑒𝑞1
𝜉𝑒𝑞1
𝜉𝑒𝑞2
𝜉𝑒𝑞3
Δ3 Δ2 Δ1Amplitude
Δ3 Δ2 Δ1
𝑇𝑒𝑞1𝑇𝑒𝑞2𝑇𝑒𝑞3
𝑆𝐷𝑒𝑞1
𝑆𝐷𝑒𝑞2
𝑆𝐷𝑒𝑞3
Sp
ectr
al D
isp
lacem
ent
(SD
)
Time Period (sec)
If Δ𝑒𝑞 𝑖,𝑅𝑜𝑜𝑓 = 𝜙𝑖,𝑅𝑜𝑜𝑓 . 𝛤𝑖 . 𝑆𝐷𝑒𝑞 𝑖 = Δ𝑖𝑛𝑖𝑡𝑖𝑎𝑙OK, Otherwise again pick up equivalent properties
corresponding to Δ𝑖 = Δ𝑒𝑞 𝑖,𝑅𝑜𝑜𝑓
𝜉𝑒𝑞3
Initial Viscous Damping
Initial Time Period
𝜉𝑒𝑞2
For 𝜉𝑒𝑞1
Determine Spectral Displacement for each Significant
Mode at Equivalent Properties
…
Equivalent Linear Properties vs. Amplitude Relationships
Mode 1
Mode 2Mode 3
Mode 1
Mode 2Mode 3