evaluation of nonlinear seismic demands of high-rise rc

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


… … … 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


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


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

+


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


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.


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


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


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”


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


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


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


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)


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


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 2

44-story Building

Mode 2



19-story Building


Mode 3

44-story Building

Mode 3



The Cyclic Behavior of

Case Study Buildings

20

20

20


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



-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)


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



-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.02 -0.01 0 0.01 0.02



Mode 1 (B2)Mode 1 (B1)

Cyclic pushover

curve

Idealized SDF

system

Ground Motion Set 4



-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)


-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


-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



Cyclic pushover

curve

Idealized SDF

system

Ground Motion Set 4



-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)


-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


-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.003 -0.001 0.001 0.003



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)


Roof Drift Ratio ( ) Roof Drift Ratio ( ) Roof Drift Ratio ( )







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.


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.


Nonlinear Responses


Strong Direction

Ground Motion Set 4


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


Nonlinear Responses


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


Nonlinear Responses


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


An Extended Displacement Coefficient Method (EDCM)

(or A Simplified Modal Pushover Analysis Procedure)


+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

)


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)


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)


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



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



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



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)



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)


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)



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


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


Series3

Series4

0

5

10

15

20

25

30

35

40

45

0 10 20 30


Series3

Series4

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40


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


EDCM – Individual Modal Demands – Ground Motion Set 4

0

5

10

15

20

25

30

35

40

45

0 20 40 60


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


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.


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


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


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


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


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)

.

.

.


“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)


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


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


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


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”?


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


Mode 3

Mode 1

Mode 2

0

5

10

15

20

25

30

35

0 10 20 30 40


0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20


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)


Ground

Motion Set 1

Ground

Motion Set 2

Ground

Motion Set 3

Mode 1 Mode 2 Mode 3




Performance of the MRSA

Procedure – Individual

Modal Demands

B3


0

5

10

15

20

25

30

35

40

45

0 200 400 600 800

No

. o

f S

tori

es

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

Overall Performance of the MRSA Procedure – B3 – Ground Motion Set 1

NLRHA

MRSA

Standard RSA

(Cd = 4, R = 4.5)


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)

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


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


• 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


• 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


Thank you for your attention

evaluation of nonlinear seismic demands of high-rise rc

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