a modified response spectrum analysis procedure...

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


From where it all started …

Lord Rayleigh (John William Strutt, 1842 - 1919)Joseph-Louis Lagrange (1736 - 1830)D’Alembert (1717–1783)


Entrance in Earthquake Engineering

Arturo Danusso (1880 - 1968) Maurice Anthony Biot (1905 - 1985)Theodore von Karman (1881 - 1963)


The Standard RSA Procedure

Raw W. Clough (1920 - 2016) Edward L. Wilson (Born 1931)Edward L. Wilson (Born 1931)

On the Standardization …

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”


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)


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


Is it justified to “equally” modify the response of each vibration mode?

Lets separate them and check one-by-one

R ???

Our Approach


The Equivalent Single-degree-of-freedom (SDF) System

m, ξi


The Uncoupled Modal Response History Analysis

≅+ + + + + + …

3 Case study Buildings

44 Story 33 Story 20 Story


Each Lollipop is Different …

≅+ + + + + + …


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 (∆ )


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


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



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


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


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)?

Natural Period Elongation


≅

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


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


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


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


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


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


0

5

10

15

20

25

30

35

40

45

0 5 10 15 20


Mode 1

UMRHA

MRSA

Standard RSA (R = 4.5)

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40


Mode 2

0

5

10

15

20

25

30

35

40

45

0 5 10 15


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


0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10


Mode 1

UMRHA

MRSA


0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30


Mode 2

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20


Mode 3

No

. o

f S

torie

sPerformance of MRSA at Individual Mode Level – B04 under EQ 2



0

5

10

15

20

25

30

35

40

45

0 5 10 15 20


Mode 2

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4


Mode 3

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25


Mode 1

UMRHA

MRSA


No

. o

f S

torie

sPerformance of MRSA at Individual Mode Level – B04 under EQ 3



0

5

10

15

20

25

30

35

40

45

0 200 400 600 800

No

. o

f S

torie

s

Displacement (mm)


0

5

10

15

20

25

30

35

40

45

0 0.25 0.5 0.75 1

IDR (%)


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 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 1


0

5

10

15

20

25

30

35

40

45

0 50 100 150 200

No

. o

f S

torie

s

Displacement (mm)


0

5

10

15

20

25

30

35

40

45

0 0.05 0.1 0.15 0.2 0.25

IDR (%)


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)


NLRHA

MRSA

Standard RSA

(Cd = 4, R = 4.5)




0

5

10

15

20

25

30

35

40

45

0 250 500 750

No

. o

f S

torie

s

Displacement (mm)


0

5

10

15

20

25

30

35

40

45

0 0.15 0.3 0.45 0.6 0.75

IDR (%)


0

5

10

15

20

25

30

35

40

45

0 10 20 30

Story Shear (x106 KN)


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)




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 × Ω

No

. o

f S

torie

s

V

MM

44-story case study building (Strong Direction)


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 × Ω

44-story case study building (Strong Direction)


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


Thank you for your attention

a modified response spectrum analysis procedure...

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