Yousef Bozorgnia,Mahmoud Hachem,Kenneth Campbell

PEER GMSM Workshop,UC BerkeleyOctober 27, 2006

Attenuation of Inelastic Spectra and Its Applications

Background: Selected Ground Motions

3122 Horizontal records

64 Worldwide earthquakes

Magnitude: 4.3 - 7.9

Distance range: 0 – 200 km

Computation of inelastic spectra

Inelastic spectra were computed for 3122 horizontal records

For periods: 0.02 to 10 sec

1999 Duzce EQ (M W 7.1), Turkey, Duzce Station, H GM

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fy /

W

= 1

= 2

= 4

= 6

= 8

D eform ation

Fy

uy um ax

Fe

Elastic response

Fs

Inelastic response

Force

Fs

= F

e / R

Fy

= F

e / R

d

ue

1999 Duzce EQ (M W 7.1), Turkey, Duzce Station, H GM

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fy /

W

= 1

= 2

= 4

= 6

= 8

D eform ation

Fy

uy um ax

Fe

Elastic response

Fs

Inelastic response

Force

Fs

= F

e / R

Fy

= F

e / R

d

ue

Computed parameters: Strength ductility displacement energy,…

Inelastic spectra for: Constant ductility; Constant strength; Constant damage index

Overall: > 1,240,000 nonlinear runs

Computation of inelastic spectra

Attenuation of Inelastic Spectra

Having computed: Inelastic spectra: 3122 (records) * 5 (ductility ratios),

and Damage spectra: 3122 (records) * 5 (damage index

values)

For a given value of ductility or damage index

Nonlinear regression analyses were performed to correlate inelastic response to Magnitude Distance to fault Soil condition Style of faulting …

Attenuation of Inelastic Spectra -- Model

3 6 7( ) ( )RV F Nf F c F f H c F

1 2 3 4 5 6ln ( ) ( ) ( ) ( ) ( ) ( )Y f M f R f F f HW f S f D

0 1

1 0 1 2

0 1 2 2

5.5

( ) ( 5.5) 5.5 6.5

( 5.5) ' ( 6.5) 6.5

c c M M

f M c c M c M M

c c M c M c M M

30 309 2 30 1

1 1

5

309 2 30 1

1

ln ln ln

( )

( ) ln

n

s sr r s

ss

V Vc k PGA c PGA c V k

k kf S

Vc k n V k

k

2 22 3 4 5( ) ( ) ln rupf R c c M r c

4 8( ) ( ) ( ) ( )RV HW HW HWf HW c F f R f M f H

10

6

3 4

( 1) 1

( ) 0 1 3

exp 10 exp 3.333 exp 0.75 exp 0.25 3

c D D

f D D

k D k D D

1 0

( )0

jb

rup jbHWjb

rup

r

r rf Rr

r

0 6.0

6.0( ) 6.0 6.5

0.51 6.5

HW

M

Mf M M

M

0 20( ) 20

2020

HW

Hf H H

H

Attenuation of Inelastic Spectra:For Different Ductility Ratios

Strike-Slip Fault, M w=7.5, T =0.2, Vs30=760, D=1

0.1 1.0 10.0 100.0

Distance to Fault Rupture (km)

0.01

0.1

1

Fy /

W

= 1

= 2

= 4

= 6

= 8

Strike-Slip Fault, M w=7.5, T =1.0, Vs30=760, D=1

0.1 1.0 10.0 100.0

Distance to Fault Rupture (km)

0.01

0.1

1

Fy /

W

= 1

= 2

= 4

= 6

= 8

Attenuation of Inelastic Spectra:For Different Magnitude

Strike-Slip Fault, = 4, T =0.2, V s30=760, D=1

0.1 1.0 10.0 100.0

Distance to Fault Rupture (km)

0.01

0.1

1

Fy /

W

M W = 7.5

M W = 6.5

M W = 5.5

M W = 4.5

Strike-Slip Fault, = 4, T =1.0, V s30=760, D=1

0.1 1.0 10.0 100.0

Distance to Fault Rupture (km)

0.01

0.1

Fy /

W

M W = 7.5

M W = 6.5

M W = 5.5

M W = 4.5

Mag Saturation

Attenuation of Inelastic Spectra

Strike-Slip Fault, M w=7.5, rrup=0.1, Vs30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

Fy /

W

= 1

= 2

= 4

= 6

= 8

C&B NGA Elastic

Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

Cy

= F

y /

W

= 1

= 2

= 4

= 6

= 8

C&B NGA Elastic

Attenuation of Elastic Spectrum Becomes a Special Case

Damage Spectra (Bozorgnia & Bertero, 2003)

Normalized Ductility Normalized Hysteretic Energy

DI1= (- e)/(mon-1) (1-1) + 1 EH/EHmon

DI=0, if Elastic response DI=1, if it reaches deformation capacity under monotonic lateral deformationOther performance states fall between DI=0,1

Damage Spectra: An example

D eform ation

Fy

uy um ax

Fe

Elastic response

Fs

Inelastic response

Force

Fs

= F

e / R

Fy

= F

e / R

d

ue

1978 Tabas EQ (M W 7.35), Iran, Tabas Station, HGM

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fy /

W

DI = 0.0

DI = 0.2

DI = 0.4

DI = 0.6

DI = 0.8

DI=0, Elastic spectrum

Attenuation of Damage Spectra

Strike-Slip Fault, M w=7.5, rrup=0.1, Vs30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

Fy /

W

DI= 0.0

DI= 0.2

DI= 0.4

DI= 0.6

DI= 0.8

Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

Fy /

W

DI= 0.0

DI= 0.2

DI= 0.4

DI= 0.6

DI= 0.8

DI=0, Elastic spectrum

Displacement Inelastic Spectra

DI= 0.4, =0.3,Strike-Slip Fault, M W=6.5, V s30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

5.0

10.0

15.0

20.0

25.0

um

ax

( c

m )

rrup= 0.1 km

rrup= 10 km

rrup= 40 km

rrup= 200 km

DI = 0.4, =0.3, Strike-Slip Fault, rrup=10.0, Vs30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

5.0

10.0

15.0

20.0

25.0

um

ax

( c

m )

M w= 7.5

M w= 6.5

M w= 5.5

M w= 4.5

Use of Inelastic Spectra for GMSM

1. Simple scaling using attenuation of inelastic spectra

2. Inelastic spectrum matching

Scaling of Ground Motion

Scaling Law of Inelastic Spectra:Bertero, Mahin, Herrera (August 1976)

For given ductility and period, If you want to scale “Yield strength” Cy

by a factor of λ You will have to scale time history by

scale factor λ

Selection and Scaling of THs: Step1

Select: Target Performance Level Select ductility; or Damage Index, or

…

Select: period

Select: site of the structure Mag, Style of Faulting, Rrup, Vs30, …

Step2: Initial Selection of Time Series

Various options … M,R,… bin M,R,ductility,… bin M,R,DI,… bin …

Example

Attenuation of Inelastic Spectra

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4

Period (sec)

Cy

= F

y /

W mu = 1

mu = 2

mu = 4

mu = 6

Inelastic Spectrum Matching

Given Inelastic Design Spectrum (e.g., attenuation of inelastic spectra)

Strike-Slip Fault, M w=7.5, rrup=10.0, V s30=760, D=1

0.0 1.0 2.0 3.0 4.0

Period (sec)

0.0

0.2

0.4

0.6

0.8

1.0

Cy

= F

y /

W

= 1

= 2

= 4

= 6

= 8

C&B NGA Elastic

And a Record

Inelastic Spectrum, Manjil EQ, Abbar Station, Ductility =4

00.10.20.30.40.50.60.70.8

0 1 2 3 4Period (sec)

Cy

(=F

y /

W )

Abbar_0

Is it possible to modify the record to match the Inelastic Spectrum?

Inelastic Spectra, Ductility =4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4Period (sec)

Cy

(=F

y /

W )

Target

Abbar_0

Answer: Yes

Inelastic Spectra; Ductility=4

Ground Motions

0 10 20 30 40 50-0.5

0

0.5

0 10 20 30 40 50

-0.2

0

0.2

Initial

Final

It is relatively fast …

Thank You!