Fawad A. Najam A Modified Response Spectrum Analysis...

A Modified Response Spectrum Analysis Procedure (MRSA) to Determine the Nonlinear Seismic Demands

of Tall Buildings

Fawad A. NajamPennung WarnitchaiAsian Institute of Technology (AIT), ThailandEmail: [email protected]

2Design of Tall Buildings: Trends and Advancements for Structural Performance (24 – 25 April 2017, NUST, Islamabad)

The Earthquake Problem

RockFault

Soil

Site

Site ResponseGround motion

can be amplified

by soil

AttenuationSeismic waves lengthen

and diminish in strength

as they travel away from

the ruptured fault.

Future ground shaking

Building

Linear/Nonlinear Analysis

Model

Characterization of Seismic

Ground Motions

Estimation of Linear/Nonlinear

Seismic Demands

• Global-level Responses

• Inter-story Responses

• Component-level Responses

… … … Simplified Estimation of

Seismic Demands

Simplified Structural Models

Simplified Loading

Any combination of

simplified approaches

Simplified Seismic Analysis

Procedures

Triangular

Modal Expansion

… … …

… … …


Dynamic Response of Structures – The Concept of Vibration Mode Shapes

The Equivalent Lateral Force

Procedure (LSPs)

Seismic

Loading

Structural Models

Detailed 3D

Linear Model

Linear Elastic

MDF Model

Linear Elastic

SDF Model

Static Single Lateral

Load Vector

Static Multiple Lateral

Load Vectors

Ground Motions

The Linear Response

History Analysis (LRHA)

Procedure

(Direct Integration)

Modal Analysis, Spectral Analysis,

Standard Response Spectrum Analysis

(RSA) Procedure

Relative Uncertainty

Low

High

The World of Linear Elastic Seismic

Analysis of Structures

or

The NLRHA Procedure

using Simplified SDF

Model

The Nonlinear Static

Procedures (NSPs)

Seismic

Loading

Structural Models

Detailed 3D

Nonlinear Model

Equivalent

Nonlinear MDF

Model

Equivalent

Nonlinear SDF

Model

Static Single Lateral

Load Vector

Static Multiple Lateral

Load Vectors

Ground Acceleration

Records

The Detailed Nonlinear

Response History Analysis

(NLRHA) Procedure

The Multi-mode Pushover

Analysis Procedures

The NLRHA Procedure

using Simplified MDF

Model

Relative Uncertainty

Low

High

The World of Nonlinear Seismic

Analysis of Structures


Linear Elastic Model

Determine Modal Properties

𝑇𝑖, 𝜙𝑖, 𝛤𝑖

𝑇1𝑇2𝑇3

𝑆𝐴1

𝑆𝐴2

𝑆𝐴3

Spe

ctr

al A

ccele

ratio

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)

𝑉𝐷𝑒𝑠𝑖𝑔𝑛 =(𝑉𝑏1 )

2 + (𝑉𝑏2 )2 + (𝑉𝑏3 )

2 + …

𝑅

Problem

• Different modes undergoes different levels of

nonlinearity.

• Reducing elastic responses of every mode by the

same R factor is not correct.

• Several ‘modified’ RSA procedures have been

proposed—using different R factor for different mode.

𝑀𝐷𝑒𝑠𝑖𝑔𝑛 =(𝑀𝑏1 )

2 + (𝑀𝑏2 )2 + (𝑀𝑏3 )

2 + …

𝑅

∆ =𝐶𝑑 (∆𝑒𝑙1)

2 + (∆𝑒𝑙2)2 + (∆𝑒𝑙3)

2 + …

𝑅


0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60

The Original Intent of R

• Basic Idea:• A structure can be economically designed for a “fraction”

of the estimated elastic seismic design forces, while maintaining the basic life safety performance objective.

• The intent of R is to simplify the structural design forces such that only linearly elastic static analysis is needed for most building design.

• “R” factor can be traced back to a “K” factor which appeared in the first edition of the Blue Book (SEAOC Recommended Lateral Force Requirements and Commentary) in 1959.

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 “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


What “R” to use?

Structural Type and MaterialUS West

CoastJapan

New Zealand**

Europe

Concrete Frame 8 1.8 – 3.3 9 5.85

Concrete Structural Wall 5 1.8 – 3.3 7.5 4.4

Steel Frame 8 2.0 – 4.0 9 6.3

Steel Eccentrically Braced Frame 8 2.0 – 4.0 9 6.0

Masonry Walls 3.5 - 6 3.0

Timber Structural Walls - 2.0 – 4.0 6 5.0

Pre-stressed Wall 1.5 - - -

Dual Wall/Frame 8 1.8 – 3.3 6 5.85

Bridges 3 – 4 3.0 6 3.5

**Sp factor of 0.67 incorporated

Source: “Direct Displacement-based Design” by Priestley MJN et al., 2007


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

Lets separate them and check one-by-one

R ???

What “R” to use?

What actually happens to the structure during the time history analysis?

Our Approach


The Concept of Modal Decomposition


Modal Decomposition of Nonlinear Seismic Responses

≅+ +…

Mode 1Mode 2

Mode 3

+

A Detailed 3D Elastic

Structural Model

Classical Modal Analysis

The Uncoupled Modal Response History Analysis (UMRHA)

F

D

F

D

F

D

NLNL NL

A Detailed 3D Inelastic

Structural Model

≅

+ + …

Mode 1Mode 2

Mode 3

+


The Uncoupled Modal Response History Analysis (UMRHA) Procedure

≅+ + + + + + …


Full 3D Nonlinear MDF Model

𝐹𝑛

𝐹1

.

.

.

Monotonic Pushover Analysis

𝑥1𝑟

𝑉𝑏1

Determination of Pushover Curve

𝑉𝑏1

𝑥1𝑟

Idealization of Pushover Curve

𝐹1

𝐷1

Equivalent SDF System

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


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

Cyclic Pushover Analysis

Base Shear (V)

Roof Drift (Δ)

V

Δ


-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

No

rmaliz

ed

Base S

he

ar

()

Roof Drift ( )

Reversed-cyclic Pushover Analysis (Solid Line)Response of an equivalent SDF system under a ground motion

Mapping the Cyclic Response to an Equivalent SDF System


-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

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.003 -0.002 -0.001 0 0.001 0.002 0.003

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

The CONCLUSION – Each Mode is Different …

Mode 1Mode 2

Mode 3

Normalized Base Shear (𝑽𝒃𝟏/𝑾) 𝑽𝒃𝟐/𝑾 𝑽𝒃𝟑/𝑾

Roof Drift (𝒙𝟏𝒓/𝑯) Roof Drift (𝒙𝟐

𝒓/𝑯) Roof Drift (𝒙𝟑𝒓/𝑯)

Cyclic pushover curve

Idealized SDF system


The CONCLUSION – Each Lollipop is Different …

≅+ + + + + + …


From UMRHA to The Modified Response Spectrum Analysis (MRSA)


What Happens when a System Starts Experiencing Nonlinearity?

k

Natural Period Elongation

Additional Hysteretic Damping

Natural Period Elongation

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


𝑇2

𝑆𝐴2

Spectr

al A

ccele

ratio

n (

SA

)

Time Period (sec)

𝑇1

𝑆𝐴1

Spectr

al A

ccele

ratio

n (

SA

)

Time Period (sec)

The Basic Concept of Modified Response Spectrum Analysis (MRSA)

Linear Elastic Structure

≅ + + + …

Mode 1Mode 2

Mode 3

F

D

F

D

F

D

𝑇3

𝑆𝐴3

Spectr

al A

ccele

ratio

n (

SA

)

Time Period (sec)

Spectr

al A

ccele

ratio

n (

SA

)

Time Period (sec)

For Initial

Viscous Damping


≅

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

Evaluation of the MRSA Procedure - 3 Case study Buildings

44 StoryB1

33 StoryB2

20 StoryB3

• 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 configuration irregular features commonly found in typical tall buildings, such as podium and non-symmetrical arrangement of RC walls, etc.


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

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4

Sp

ectr

al A

cce

lera

tio

n (

g)

Period (sec)

7.5%-damped

Mean Spectrum

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Sp

ectr

al A

cce

lera

tio

n (

g)

Period (sec)

0

0.1

0.2

0.3

0 0.5 1 1.5 2 2.5 3 3.5 4

Sp

ectr

al A

cce

lera

tio

n (

g)

Period (sec)

2.5%-damped

Mean Spectrum

5%-damped

Mean Spectrum

Target

Mean of Matched GMs


7.5%-damped

Mean Spectrum

5%-damped

Mean Spectrum

2.5%-damped

Mean Spectrum

7.5%-damped

Mean Spectrum

5%-damped

Mean Spectrum2.5%-damped

Mean Spectrum

Set 2

Set 3

Set 1


Ground Motions

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

Sp

ectr

al A

cce

lera

tio

n (

g)

Period (sec)

5%-damped Target Response Spectrum

Mean of Matched Ground Motions


𝜉 = 2.5%

𝜉 = 10%

𝜉 = 5%

Typical Code Target Spectrum

and Matched Ground Motions

– Set 4

Set 4


MVLEM for RC WallsLumped Fibers for RC ColumnsMasonry Infill Wall Model (FEMA 356)

Nonlinear Modeling of Case Study Buildings

Level m

Level m-1

Rigid Link

Concrete and Steel fibers

Linear shear springLinear-elastic frame

element (shear deformation excluded)

Fiber section element (concrete and steel fibers)

Linear shear spring

-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 tensionN

orm

aliz

ed B

ase S

hear

(𝑉𝑏1/𝑊

)

Roof Drift (𝑥𝑖𝑟/𝐻)

Strong (𝑥) Direction


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/𝑊) 𝑉𝑏/𝑊 𝑉𝑏/𝑊

Roof Drift (∆/𝐻) Roof Drift (∆/𝐻) Roof Drift (∆/𝐻)

20-story 33-story 44-story


UMRHA vs. NLRHA

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50


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


Shear Wall Cracking

UHS-Matched Ground Motion 0.2 sec CMS-Matched Ground Motion 3 sec CMS-Matched Ground Motion

25% of Cracking Strain



Already Cracked

Set 1 Set 2 Set 3


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)

(a) 𝑇𝑠𝑒𝑐,𝑖/𝑇𝑖 vs. Roof Drift Ratio (𝑥𝑖𝑟/𝐻)

“Equivalent Linear” Properties


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


0

5

10

15

20

25

30

35

40

45

0 5 10 15 20


Mode 1

UMRHAMRSAStandard 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 – EQ 1


0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10


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


Mode 2

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20


Mode 3

No. of S

tories




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

Standard RSA (R = 4.5)

No. of S

tories




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

tori

es

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 4)

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)

NLRHA Mean

MRSA

Standard RSA

Standard RSA × ΩNo

. o

f S

torie

s MM

44-story case study building (Strong Direction)


Local Response – Shear Walls (EQ Set 4)

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

-15 -10 -5 0 5 10 15

Shear (x106 N)

No

. o

f S

torie

s V M

NLRHA Mean

MRSA

Standard RSA

Standard RSA × Ω



Local Response – Axial Strain in Shear Walls (EQ Set 4)

0

5

10

15

20

25

30

35

40

45

-400 -300 -200 -100 0 100 200 300 400

Axial Strain (x106 mm/mm)

0

5

10

15

20

25

30

35

40

45

-1000-750 -500 -250 0 250 500 750 1000

Axial Strain (x106 mm/mm)

Tensile

Cracking

Tensile

Cracking

No

. o

f S

torie

s

NLRHA Mean

MRSA

Standard RSA

Standard RSA × Ω



Seismic Analysis Practice for Structural Design - The Case of Pakistan

Source: Zaman and Warnitchai (2016)


How Safe are Our Buildings Against Earthquakes?

Peak Ground Acceleration (g)

Source: Durrani et al.,

Abbottabad, 8th October 2005


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Period (sec)

SD

SC

SB

SA

Elastic Spectrum of EW

Component

Abbottabad, 8th October 2005

How much we understand?

Peak Ground Acceleration



Spectral Acceleration at 0.2 sec


Spectral Acceleration at 1 sec


Thank you for your attention

Fawad A. Najam A Modified Response Spectrum Analysis...

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