4.seismic ground response or soil “amplification” of...

4.Seismic Ground Response

orsoil “amplification” of seismic

ground motions

G. BouckovalasProfessor N.T.U.A..

July 2010.

Suggested Reading:Suggested Reading:

Steven Kramer: Chapter 7 (7.1 & 7.2)George Gazetas: Chapter 4 (4.2)

SHAKE ’92 – Users’ Manual

GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.1

The examples which follow come from real seismic events and may

help fro a first qualitative as well as quantitative evaluation of soil

effects on recorded seismic ground motions. Thus, keep notes with

regard to the:

Α. Soil “amplification” coefficientΒ. Important soil and seismic motion parametersC. Frequency dependence of soil amplification effects

4.1 EXAMPLES FROM (REAL) CASE HISTORIES

San Francisco 1957 . . . .

Variation of peak ground acceleration and elastic response spectrain connection with local soil conditions at recording stations . . .GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.2

San Francisco 1989 . . . .

Recording atoutcropping ROCK

Recording at the surface of earth fill on Bay mud

75 cm/sec2

170 cm/sec2

Soil “amplification” or “de-amplification” ?


Soil “amplification” or “de-amplification” ?

Is this a pure soil effect?


Kobe, Japan 1995

stiff soil, amax=0.82g

loose fill, amax=0.32g

Αthens, Greece 1999 . . . .

Demokritus: rock, amax=0.06g

Chalandri: soil, amax=0.19g


Moderate damage

Slight or no damage

Severe damageor collapse

N

S

N

S

STIFF SOIL AMPLIFICATION at Ano Liosia municipality

Αthens, Greece 1999 . . . .

0

25

50

75

100

(%)

of

bu

ildin

gs

BoreholeCHT2

DH5

80

60

40

20

0

DE

PT

H (

m)

Vs (m/sec)

_ Vs,30 =697 m/sec

360 7600 1300

_ Vs,30 =613

_ Vs,30 =545 _

Vs,30 =500

_ Vs,30 =496

N S

0 1300 0 1300 0 1300 0 1300

Cross-section N-S

Conglomerates Clayey MarlGEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.6

Mexico City 1985

geology thickness of soil deposits

Typical recordings on the surface of SOIL & ROCK

Fourierde-composition

Fouriercomposition

Simplified mechanism of SOIL effects

Fourier Spectrum

Fourier Spectrum

Transfer function

frequency

time

time


Α. Soil “amplification” coefficient:

Β. Important soil and seismic motion parameters:

C. Frequency dependence of soil amplification effects:

Initial Conclusions :

(0.40-3.20) 0.6-2.0

soil & excitation characteristics

amax & Sa

Quite often,

the effect of a few (tens of) meters of soft soilis larger and more significant thanthan the effect a few kilometers of earth crust….

Quite often,

the effect of a few (tens of) meters of soft soilis larger and more significant thanthan the effect a few kilometers of earth crust….


Seismic Codes: ΕΑΚ-2000 ………

π.χ.Κατηγορία Β: Εντόνως αποσαθρωμένα βραχώδη ή εδάφη που απόμηχανική άποψη μπορούν να εξομοιωθούν με κοκκώδη. Στρώσειςκοκκώδους υλικού μέσης πυκνότητας (και) πάχους μεγαλύτερουτων 5μ, ή μεγάλης πυκνότητας (και) πάχους μεγαλύτερου των 70μ.

Vs,30=180-360m/s

NSPT=15-50

Cu=70-250KPa

Soil Soil ParametersParameters

EE

DD

Deep deposits of dense or medium-dense sand, gravel or stiff clay with thickness from several tens to many hundreds of m

CC

BB

AA

DescriptionDescription

0.05

0.10

0.10

0.05

0.05

M<5,5M<5,5

0.15

0.20

0.20

0.15

0.15

M>5,5M>5,5

TTBB

1.6

1.8

1.5

1.35

1.0

M<5,5M<5,5

1.4

1.35

1.15

1.2

1.0

M>5,5M>5,5

SS TTCC

0.25

0.30

0.25

0.25

0.25

M<5,5M<5,5

0.5EE

0.8DD

0.6CC

0.5BB

0.4AA

M>5,5M>5,5

0 0.5 1 1.5T (s)

0

1

2

3

Se

/ (S

*ag)

A

B,EC

D

M>5.5

Seismic Codes: EC-8 ………


0 0.5 1 1.5T (s)

0

1

2

3

Se /

(S*a

g)

A,B,C,E

D

Vs,30=180-360m/s

NSPT=15-50

Cu=70-250KPa

Soil Soil ParametersParameters

EE

DD


CC

BB

AA

DescriptionDescription

0.05

0.10

0.10

0.05

0.05

M<5,5M<5,5

0.15

0.20

0.20

0.15

0.15

M>5,5M>5,5

TTBB

1.6

1.8

1.5

1.35

1.0

M<5,5M<5,5

1.4

1.35

1.15

1.2

1.0

M>5,5M>5,5

SS TTCC

0.25

0.30

0.25

0.25

0.25

M<5,5M<5,5

0.5EE

0.8DD

0.6CC

0.5BB

0.4AA

M>5,5M>5,5

M<5.5

Seismic Codes: EC-8 ………


I. Analytical (harmonic excitation, uniform & elastic soil, ..)

II. Empirical (based on geological and seismological input)

III. Numerical (SHAKE, DESRA, …)

In principle, analysis-prediction of soil effects is possible with one of the three following methods (with greatly different complexity, as well as, accuracy):

accu

racy

simplicity


Uniform elastic soil on rigid bedrock

Uniform visco-elastic soil on rigid bedrock

Uniform visco-elastic soil on flexible bedrock

Non-uniform visco-elastic soil on flexible bedrock

4.3 ANALYTICAL METHODS4.3 ANALYTICAL METHODS

Definitions :

( )

s b rs b

GV

ρ

4H 4HT ,T ήT

V V

ωk wave number

V

=

= =

=

Vs, ρs, ξs HVb, ρb, ξb(ή Vr, ρr, ξr)

Elastic soil no RIGID bedrock


( ) ( )kztikzti BeAeu −+ += ωω

stationary wave with amplitude

kH

kzukzAu oo cos

coscos2 ==

tioeuu ω=

( )

( )kzti

kzti

Be

Ae−

+

ω

ω

z H( ) ( )

( )

z 0

i ωt kz i ωt kz

z 0

iωt

ikz ikziωt iωt

Boundary condition : τ 0

uG 0 Gik Ae Be 0

z

for z 0 : Gik A B e 0 A B

e ethus : u 2 Ae 2 Ae cos kz

2

=

+ −

=

−

=

∂ ⎡ ⎤= ⇒ + =⎣ ⎦∂

= − = ⇒ =

+= =

iωtz H o

oo

Boundary condition : u u e

uthus : u 2 Acos kH ή A

2 cos kH

= =

= =

Amplification factor: ( )

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

=

=

=

⎥⎦⎤

⎢⎣⎡ −=

=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛==

ΒΓ

=

3,2

5cos

2,2

3cos

1,2

cos

2

122

cos2

,2,1,12

2cos

1

cos

1

.

1

nH

z

nH

z

nH

z

A

uή

H

zn

A

u

nnTT

TTkHί

ίF

exc

soil

exc

soil

π

π

π

π

πνησηκνησηκω

K

L

Resonance:

Modal shapes:


|F1|

0 1 3 5 7 Ts/Texc

n=1

n=3

n=2

1

00 1-1

z/H

u/2A

( )

( )

1

.

1 1

coscos

2

2 1, 1, 2,

cos 2 12 2

cos , 12

3cos , 2

2 25

cos , 32

soil

exc

soil

exc

motionF

motion kH T

T

T n nT

u zn

A H

zn

Hu z

ή nA H

zn

H

ωπ

π

π

π

π

Γ= = =

Β ⎛ ⎞⎜⎝ ⎠

= − =

⎡ ⎤= −⎢ ⎥⎣ ⎦⎧ =⎪⎪⎪= =⎨⎪⎪

=⎪⎩

L

K

Amplification factor:

συντονισμός:

ιδιομορφές:

|F1|

0 1 3 5 7 Ts/Texc

Due to the RIGID bedrock assumption(motion Α) = (motion Β)And consequently

1

motion motionF ( )

ί motion

Γ Γωκ νηση Β Α

= =


Definitions for elastic soil (ξ=0)

( )

s b rs b

GV

ρ

4H 4HT ,T ή T

V V

ωk wave number

V

=

= =

=

Vs, ρs, ξs HVb, ρb, ξb

(ή Vr, ρr, ξr)

Visco-elastic soil on RIGID bedrock

Definitions for visco-elastic soil (ξ≠0)

( )

( )

( ) ( )

( ) ( )

G G i

GV V i

H HT i T i

V V

k i k iV V

= +

= ≈ + ≤

= ≈ − = −

= ≈ − = −

*

**

**

**

1 2

1 0,30

4 41 1

1 1

ξ

ξ για ξρ

ξ ξ

ω ω ξ ξ

( ) ( )[ ] ( )kHikHikHHkF

ξξω

−=

−≅=

cos

1

1cos

1

cos

1*2

According to the “correspondence principle”, you may use the solution for elastic soil, with the real soil variables replacedby the corresponding complex variables. Thus,

Amplification factor

( ) ( )0cossinhcoscos 2222 →+≈+=± yyxyxiyx

( )( )

2 22

1/

cossF where V

kH kHω κ ω

ξ≈ =

+

or, given that:

( ) ( )

( )

2

2 1max

2 1

2 1 2 12

soil

exc

Fn

when

TkH n or n

T

ωπ ξ

π

=−

≈ − = −GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.15

Elastic soil on RIGID bedrock

Visco-elastic soil on RIGID bedrock

Uniform visco-elastic soil on FLEXIBLE bedrock

( )

( ) ( )b

bb

s

ssbssb

tissss

s

sss

z

uG

z

uGuHuHzz

ezkAuz

uGz

∂∂

=∂∂

===

=→=∂∂

==

&0:,0

cos20:0 * ωτ L

( ) ( )[ ]( ) ( )[ ]⎪

⎩

⎪⎨

⎧

++−=

−++=⇒

−

−

HikHiksb

HikHiksb

ss

ss

eaeaAB

eaeaAA

**

**

**

**

112

1

112

1

b

s

b

s

bb

ss

bb

ss

i

ia

i

i

V

V

V

Va

ξξ

ξξ

ρρ

ρρ

++

=++

==1

1

1

1*

**

Boundary conditions:

ρs,Vs

ρb,Vb

zs

zb

[ ] tisss ezkAu

s

ω*cos2=

H

( )bb zktibeA

*+ω

( )bb zktibeB

*−ω


( ) ( )( )

( ) ( ) ( ) ( )[ ])(F

HkHkcos

1

Hkcos

1F

BA

A2

0u

0u

u

uF

22

12

sss

2*

s

3

bb

s

b

s

B

A3

ωξ

ω

ω

=+

≈=

⇒+

=== L

( )

( ) ( ) ( ) ( ) ( )[ ] 21

2

sss

2*

s

*

s

*

s

4

b

sA4

HkHkcos

1

HksiniaHkcos

1F

A2

A2

u

uF

ξω

ωΓ

+≈

+=

⇒== L

για άκαμπτο βράχο, Vb ∞, as 0, ξs ξs, έτσι:

( )( ) ( )[ ]

( )ωξ

ω 32

12

sss

24 F

HkHkcos

1F =

+=

bb

sss

s

.excsss

V

Va

T

Ta

2

ό

ρρπ

ξξ

που

=

+=

However, in practice it is more important to know the ratio

UA/UΓ:


1

exc

ssss

exc

s

s

s

T

Ta

2

T

T

2V

HHk

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

==

πξξ

πω

Problem parameters:

( ) ( ) ( ) ( ) ( )[ ] 21

2

sss

2*

s

*

s

*

s

4

HkHkcos

1

HksiniaHkcos

1F

ξω

+≈

+=

.2: ,exc s s

s s s ss b b

T Vwhere a a

T V

ρξ ξ

π ρ= + =

)T

Tή(

T

T

V

Va

TT

exc

b

S

b

bb

sss

s

excs

≈=ρρ

ξ

Important soil and excitation parameters affecting soil amplification . . .


Analysis of damage from previous earthquakes

Correlation to (surface) geology

Attenuation relationships for αmax, Vmax, ….

Correlation to shear wave velocity VS

Seismic Codes

Semi-analytical relationships

4.4 EMPIRICAL METHODS4.4 EMPIRICAL METHODS

Damage analysis from previous earthquakes:San Francisco iso-seismal mapFrom the 1906 earthquake


Correlations to (surface) geology:

Factors of average spectral amplificationAtt

enua

tion

Relat

ions

hips

for

α max, V m

ax, ….


0.05

0.10

0.10

0.05

0.05

M<5,5M<5,5

0.15

0.20

0.20

0.15

0.15

M>5,5M>5,5

TTBB

1.6

1.8

1.5

1.35

1.0

M<5,5M<5,5

1.4

1.35

1.15

1.2

1.0

M>5,5M>5,5

SS TTCC

0.25

0.30

0.25

0.25

0.25

M<5,5M<5,5

0.5EE

0.8DD

0.6CC

0.5BB

0.4AA

M>5,5M>5,5

0 0.5 1 1.5T (s)

0

1

2

3

Se /

(S*a

g)

A

B,EC

D

M>5.5

0 0.5 1 1.5T (s)

0

1

2

3S

e /

(S*a

g)

A,B,C,E

D

M<5.5

Recent Seismic Codes: π.χ. EC - 8

Parameters Ground

type Description of stratigraphic profile

Vs,30 (m/s) NSPT (blows/30cm)

cu (kPa)

A Rock or other rock-like geological formation, including at most 5 m of weaker material at the surface

> 800 _ _

B

Deposits of very dense sand, gravel, or very stiff clay, at least several tens of m in thickness, characterised by a gradual increase of mechanical properties with depth

360 – 800 > 50 > 250

C


180 – 360 15 - 50 70 - 250

D

Deposits of loose-to-medium cohesionless soil (with or without some soft cohesive layers), or of predominantly soft-to-firm cohesive soil

< 180 < 15 < 70

E

A soil profile consisting of a surface alluvium layer with Vs,30 values of type C or D and thickness varying between about 5 m and 20 m, underlain by stiffer material with Vs,30 > 800 m/s

S1

Deposits consisting – or containing a layer at least 10 m thick – of soft clays/silts with high plasticity index (PI > 40) and high water content

< 100 (indicative)

_ 10 - 20

S2

Deposits of liquefiable soils, of sensitive clays, or any other soil profile not included in types A –E or S1


0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

S

EC8

Vb=800m/s

Vb=1200m/s

Average

A B C D E

Soil Factors – Soil Categories EC 8– Effect of VS,30Soil Factors – Soil Categories EC 8– Effect of VS,30

A

B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs

,30

(m/s

ec)

Vb=800m/s

A

B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs

,30

(m/s

ec)

Vb=1200m/sVb=800 m/s Vb=1200 m/s

H (m) H (m)

Vs,

30(m

/s)

(M > 5.5)

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

S

EC8

Vb=800m/s

Vb=1200m/s

Average

A B C D E

A

B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs,

el (

m/s

ec)

Vb=800m/s

A

B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs,

el (

m/s

ec)

Vb=1200m/s

Vb=800 m/s Vb=1200 m/s

H (m) H (m)

VS

(m/s

)

(M > 5.5)GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.22

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

S

EC8

Suggested

Vb=800m/s

Vb=1200m/s

Average

A1 B C D EA2

A1

A2

A2 B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs

,el (

m/s

ec)

Vb=800m/s

A1

A2

A2 B

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

300

400

500

600

700

800

Vs

,el (

m/s

ec)

Vb=1200m/s

H (m) H (m)

VS

(m/s

)

Vb=800 m/s Vb=1200 m/s

(M > 5.5)


linear visco-elastic SEISMIC BEDROCKρb , ξb , Vb , Tb= 4H / Vb

ρs , ξs , Vs Ts= 4H / Vs

layer i: ρsi , ξsi , Vsi

H

A CC A

BB´

linear visco-elastic soil nonlinear visco-elastic soil

TeT = Te

Semi-analytical relationships (Bouckovalas & Papadimitriou, 2003)

REAL conditionsIDEALIZED conditions

linear visco-elastic SEISMIC BEDROCKρb , ξb , Vb , Tb= 4H / Vb

ρs , ξs , Vs Ts= 4H / Vs

layer i: ρsi , ξsi , Vsi

H

A CC A

BB´

linear visco-elastic soil nonlinear visco-elastic soil

REAL conditions

Semi-analytical relationships (Bouckovalas & Papadimitriou, 2003)

Soil ‘Amplification’ =motion at A

motion at C

IDEALIZED conditions


Database with more than 700 SHAKE-type analyses

0

60

120

<100

150-

200

250-

300

350-

400

450-

600

Vs = 50–700 m/s

Ts = 0.04–3.33 s

0

60

120

0.04

-0.1

0.2-

0.3

0.4-

0.5 0.6-

0.8

1.0-

1.50

Site Characteristics

Vb = 100–1000 m/s

0

200

400

103

408

550

600 65

0

750

>800

H = 3.5–240 m

0

60

120

0-5 10

-15 20

-25

30-3

545

-55

65-1

0015

0-24

0

Excitation Characteristics

abmax = 0.01–0.45g

0

100

200

300

0,01

-0,0

7

0,15

-0,2

1

0,26

-0,3

4

0,41

-0,4

5

n = 0.5–24

0

100

200 0.5

11.

52.

53.

5 44.

5 5 6

724

Step 1: Non-linear Soil Period, Ts

best fit relation:

0.1 1 2 3

TS : sec (data)

0.1

1

2

3

TS :

sec

(p

redi

ctio

ns)

over

estim

ation

by 5

0%

unde

resti

mat

ion

by 3

3%

0.03

perfectagreement( )

( ) 31

041

53301.

,

.max

,os

b

ossV

aTT +=


Step 2: ‘Amplification’ of peak ground acceleration, Aa

best fit relation:

( )

( )[ ] ( )222

22

21

1

1

eses

esa

/TTC/TT

/TTCA

+−

+=

( ) ⎥⎦

⎤⎢⎣

⎡

+=

−

n

naC b

121

1701

.max.

s

b

T

TC 5700512 .. +=

0 1 2 3 4 5 6

Ts / Te

0

0.5

1

1.5

2

Aa

DataTb/TS = 0.05 - 0.40

n = 0.5 - 2ab

max = 0.41 - 0.45

best fitrelation

Step 3: ‘Amplification’ of normalized Elastic Response Spectra, ASa*

-0.4

0

0.4

a (

g)

0 4 8 12

time (sec)

-0.4

0

0.4

a (

g)

0 0.4 0.8 1.2 1.6 2

Tstr (sec)

0

0.3

0.6

0.9

1.2

Sa (

g)

0.01 0.1 1 10

Tstr / TS

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

AS

a* =

AS

a / A

a

excitation 1

excitation 2

commonab

max= 0.3g

ASa,r*

fit relationASa,p*

*,rSaAB =1( )

*

*

,

,

pSa

rSa

A

AB

+=

12

fit relation:

( )

( )[ ] ( )222

22

21

1

1

sstrsstr

sstrSa

/TTB/TT

/TTBA

+−

+=*


Peak Spectral ‘Amplification’, ASa,p*

best fit relation:

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤+

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−+

≤⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

e

s

e

s

e

s

e

s

e

s

pSa

T

TD

T

T

T

TD

T

T

T

T

A

433181

4113181

1318010580

,.

,.

,.

*

.

,6130

5040

2790 ..

. −−

⎟⎟⎠

⎞⎜⎜⎝

⎛= n

T

TD

s

b

0 1 2 3 4 5 6 7 8

TS / Te

0

2

4

6

8

ASa,p*

Datan= 0.5 - 1

Tb/TS= 0.05-0.3ab

max=0.40-0.45g

best fitrelation

Residual Spectral ‘Amplification’, ASa,r*

0 1 2 3 4 5 6 7 8 9 10

Ts / Te

0

2

4

6

8

ASa,r*

Datan=0.5 - 1.5

Tb/TS= 0.05-0.4ab

max= 0.01-0.45g

best fitrelationbest fit

relation:

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤+

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−+

≤⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

e

s

e

s

e

s

e

s

e

s

rSa

T

TF

T

T

T

TF

T

T

T

T

A

656980

6116980

130201

,.

,.

,.

*,4060

4740

1890 ..

. −−

⎟⎟⎠

⎞⎜⎜⎝

⎛= n

T

TF

s

b


VERIFICATION CASE STUDY3 sites during the Northridge 1994 earthquake (Mw=6.7)

I-5

HansenDam

HWY 101

I-405

HWY 101

Santa Susana Mts

PacoimaReservoir

San GabrielMts

HWY 170 VerdugoMts

Santa Monica Mts

0 1 2 3 miles

N

LDF

RRS

SFY

SAN FERNANDO VALLEY

HWY 180

NorthridgeEpicenter

LDF: ‘bedrock’ site

RRSSFY

‘soil’ sites

(Gibbs et al. 1999, Cultrera et al. 1999)

Soil conditions at recording sitesLDF

Clay -Silt

Sand - Gravel

DenseSand

0 300 600 900

VS,o (m/s)

80

70

60

50

40

30

20

10

0

80

70

60

50

40

30

20

10

0

dept

h (

m)

DenseSand

RRSSilt

Gravel

Silt

Silt-SandStone

very hardsilt stone

SFY

sand -gravel

FineSand

Silt

Gravel

LDF

SFYRRS


0.01 0.1 1

Tstr (sec)

0

1

2

Saso

il (g

)

0.01 0.1 1

Tstr (sec)

0

1

2

0.01 0.1 1

Tstr (sec)

0

1

2

Approx. Relations vs Records

Num. Predictions vs Records

Approx. Relations vs Numer. Predictions

Site response at SFY (Arleta Fire Station)

Site response at RRS (Rinaldi Receiving Station)

0.01 0.1 1

Tstr (sec)

0

1

2

Sa

soil (

g)

0.01 0.1 1

Tstr (sec)

0

1

2

0.01 0.1 1

Tstr (sec)

0

1

2

Approx. Relations vs Records

Num. Predictions vs Records

Approx. Relationsvs Numer. Predictions

Approx. Relations ≅ Numerical Predictions


Homework 4.1Homework 4.1: : Soil effects in Soil effects in LefkadaLefkada, Greece, Greece (2003)(2003) earthquakeearthquake

The accompanying figures provide the basic data with regard to the recent (2003) strong motion recording in the island of Lefkada:

-Acceleration time histories and elastic response spectra (5% structural damping) from the two horizontal seismic motion recordings on the ground surface.

-Acceleration time histories and elastic response spectra (5% structural damping) for the two horizontal seismic motion recordings on the surface of the outcropping bedrock, as computed with a non-linear numerical analysis

-Soil profile at the recording site.

Using the semi-analytical relationships, COMPUTECOMPUTE the ground surface spectral accelerations, at 5-6 representative structural periods, using the seismic recordings at the outcropping bedrock as input.

Compare with the actual recordings and comment on causes of any observed differences.

(NOTE: Choose the LONG component of seismic motion for your computations)

-0.4

-0.2

0

0.2

0.4

a(g

)

0 5 10 15 20 25

Time (s)

-0.4

-0.2

0

0.2

0.4

a(g

)

0,34

0 5 10 15 20 25

Time (s)

0,42 Surface

Out.Bedrock

0,200,35

Surface

Out.Bedrock

LONG TRANS


0

0.6

1.2

1.8

Sa(

g)

0 0.4 0.8 1.2 1.6 2

Tstr (s)

0.5

1

1.5

2

2.5

AS

a

0 0.4 0.8 1.2 1.6 2

Tstr (s)

TRANSLONG

Soil profile at the recording siteSoil profile at the recording site

0 100 200 300 400 500

VS (m/sec)

30.0

CL

MARL(CL)

ML

4.0

12.0

Vb = 450m/s

SC 8.0

6.0

0 10 20 30 40 50

NSPT

30

25

20

15

10

5

0

De

pth

(m)

>50

>50

>50

>50

>50

>50

0

0.2

0.4

0.6

0.8

1

G/G

max

γ (%)

0

10

20

30

ξ (%

)

H = 24m

Seismicbedrock

12

3

12

3

CH

10-4 10-3 10-2 10-1 1 10

6.0

2.0


EQUIVALENT LINEAR

FREQUENCY DOMAIN ANALYSIS

NON-LINEAR

TIME DOMAIN ANALYSIS

4.4 NUMERICAL METHODS4.4 NUMERICAL METHODS

EQUIVALENT-LINEAR ANALYSIS(also known as complex response method in the frequency domain)

(1)

(i) hiρi, Goi, ξoi

Response

at the free ground surfaceor at theinter-face between layers

ρb, Gb (ξb=0)Excitationat the “outcropping”or at the“buried”ΥΠΟΒΑΘΡΟ GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.32

motion at layer m

( )( )

( )( )

( )( ) tihik

mhik

mmmhm

tihikm

hikmhm

timmmmom

timmom

tizikm

zikmmm

m

mmm

tizikm

zikmm

eeBeAkiG

eeBeAu

eBAkiG

eBAu

eeBeAkiGz

uG

eeBeAu

mmmm

mmmm

mmmm

mmmm

ω

ω

ω

ω

ω

ω

τ

τ

τ

**

**

**

**

**,

,

**,

,

***

−

−

−

−

−=

+=

−=

+=

−=∂∂

=

+=

u1

z1

um

zm

1

m

m+1

ρ1, V1, ξ1

ρm, Vm, ξm

ρm+1, Vm+1, ξm+1

um+1

zm+1

This numerical method is essentially based on the analytical solution for . . .

NONUNIFORM (layered) visco-elastic soil on flexible bedrock.

για zm=0:

για zm=hm:

Boundary constraints at the free ground surface

( ) ti

ti

ezkkiGA

ezkAu

BA

ω

ω

τ

τ

1*1

*1

*111

1*111

110,1

sin2

cos2

0

=

=

=⇒=

Boundary constraints between layers

( )mmmm

m

mmmm

m

hikm

hikm

mm

mmmmhmm

hikm

hikmmmhmm

eBeAGk

GkBA

eBeABAuu

**

**

*1

*1

**

11,0,1

11,0,1

−

+++++

−+++

−=−⇒=

+=+⇒=

ττ

and finally

( ) ( )

( ) ( )

* *

* *

* **1`

**

* * 1 11`

1 11 1

2 21 1

1 12 2

m m m m

m m m m

ik h ik hm m m m m

m mm

ik h ik h m mm m m m m

A A a e B a eV

ό aV

B A a e B a e

ρπουρ

−+

− + ++

= + + −=

= − + +GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.33

Sequential application, from the free ground surface (i=1) to deeper layers (i=m)

i=1 ( ) ( )( )( ) ( )

( )

* *1 1 1 1

* *1 1 1 1

* *2 1 1 1 1

* * *1 1 1 1 1 1

* *2 1 1 1 1

* * *1 1 1 1 1 1

1 11 1

2 2

cos sin

1 11 1

2 2

cos sin

ik h ik h

ik h ik h

A A a e A a e

A k h ia k h

B A a e A a e

A k h ia k h

−

−

= + + −

= +

= − + +

= −

( )( )

*2 1 1 1 1

*2 1 1 1 1

A f k h A

B g k h A

=

=

or,briefly

i=2 ( ) ( )

( ) ( )

( ) ( ) ⎥⎦⎤

⎢⎣⎡ ++−=

⎥⎦⎤

⎢⎣⎡ −++=

−++=

−

−

−

2*22

*2

2*22

*2

2*22

*2

*21

*2113

*21

*211

*22

*223

12

11

2

1

12

11

2

1

12

11

2

1

hikhik

hikhik

hikhik

eaeaaAB

eaeaaA

eaBeaAA

β

β

( )( )

* *3 2 1 1 2 2 1

* *3 2 1 1 2 2 1

,

,

A f k h k h A

B g k h k h A

=

=

or,briefly

i=mAm=Α1•fm-1(k1

*h1, k2*h2,…, km

*hm)

Bm=A1•gm-1(k1*h1, k2

*h2,…, km*hm)

Considering that Α1=Β1, the soil amplification factor becomes:

Γ Α

Β’ Β

(uB ≈ uB’)

( )

( )

( ) ( ) ( )

A 1 13

B m m m 1 m 1

A A4

B B'

224 3 b b b

u A B 2F

u A B f g

u u uF

u uu

F F ( ) cos k H k H

− −

+= = =

+ +

= = ⇒

= • +

Γ

Γ

ω

ω

ω ω ξ


The basic input data required for this type of analysis are the following:

Acceleration time history for the seismic excitation.

The elastic shear modulus Go,b and the specific mass density ρb of the seismic bedrock

The depth and the thickness of each soil layer i

The elastic shear modulus Go,i and the specific mass density ρi of each soil layer i

Experimental curves describing the variation of the shear modulus and hysteretic damping ratios with cyclic shear strain amplitude (G/Go-γ and ξ-γ curves)

±τ

±γc

1

Go1

G < Go (απομείωση)

γc

-γc

γ

τ

ΔΕ

Εελ.

.4

1

ελΕΔΕ

πξ =

• shear modulus degradation

• hysteretic energy loss

cyclic loading

earthquake


Experimental curves for the G/Go-γ and ξ-γ relations (Vucetic & Dobry, 1991)

OCR = 1-8

PI = 0

15

30

50

100

200

25

20

15

10

5

0

10-4 10-3 10-2 10-1 100 10

γc%

ξ%

OCR = 1-15 015

3050

100PI=200

1.0

0.8

0.6

0.4

0.2

0.0

10-4 10-3 10-2 10-1 100 10

GGmax

γc%

SAN

D(I

p =0) SAN

D(I p

=0)

γ (%) γ (%)

ξ(%

)

G /G m

ax

Effect of soil type (through IP ή PI)

(Ishibashi 1992)( ) ( ) ( )

( ) ( )

( ) ( ) ( )

m ,PI mo'm

max

0.492

6 1.4040.41.3

o

GK , PI

G

0.000102 n PIK , PI 0.5 1 tanh ln

0.0 PI 0

3.37 10 PI 0 PI 150.000556m , PI m 0.272 1 tanh ln exp 0.0145PI n PI

7.0 10

γγ σ

γγ

για

γιαγ

γ

−

−

=

+= +

=

× < ≤− = − − =

×

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎢ ⎥⎨ ⎬⎜ ⎟⎢ ⎝ ⎠ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

7 1.976

5 1.115

PI 15 PI 70

2.7 10 PI PI 70

για

για

−

−

< ≤

× >

⎧⎪⎪⎨⎪⎪⎩

Effect of effective consolidation stress on the G/Go-γ and ξ-γ relations


Solution Sequence

1st Step: Fourier analysis (transform) of the seismic excitation into harmonic components

2nd Step: Gl=Go,l και ξl=ξο

3rd Step: Computation of transfer functions Fi,j for each soil layer and each harmonic excitation component

4th Step: Computation of ground response for each harmonic excitation component

5th Step: Inverse Fourier analysis (transform) of the harmonic ground response components for the computation of the seismic ground response

6th Step: Computation of maximum shear strain amplitudeγmax at the middepth of each soil layer

7th Step: Computation of the shear modulus G and damping ratio ξvalues which correspond to 2/3 γmax.


EQUIVALENT LINEAR

FREQUENCY DOMAIN ANALYSIS

NON-LINEAR

TIME DOMAIN ANALYSIS

4.4 NUMERICAL METHODS4.4 NUMERICAL METHODS

The shear stress-strain relationship (τ-γ) for monotonic and cyclic loading of each soil layer

NON-LINEAR ANALYSIS(time domain analysis)

1

ihi ρi, Goi, ξoi

excitation

response

The basic input data include:

Acceleration time history for the seismic excitation.

The elastic shear modulus Go,b and the specific mass density ρbof the seismic bedrock

The depth and the thickness of each soil layer i

The elastic shear modulus Go,i and the specific mass density ρiof each soil layer i


The layered soil profile is discretized and simulated as a system of lamped masses and visco-elastic springs . . .

bMX KX MU+ = −&& &&

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nM

M

M

MO

2

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−+−

−+−−

=

O3

3322

2211

11

00

0

0

00

K

KKKK

KKKK

KK

Kwith

The layered soil profile is discretized and simulated as a system of lamped masses and visco-elastic springs . . .


Regardless of the solution algorithm, it is mandatory to define the shear stress-strain relationship (τ-γ) which

controls the dynamic response (loading-unloading-reloading) of each soil layer.

Given that G=f(τ ή γ), it is evident that also Κ=f(U). Hence, we deal with a non-linear differential equation, which has to be solved incrementally, in very small time steps (small enough to ensure convergence).

Solution of the previous differential equation is achieved with time integration, for given initial

conditions and a given time history of base excitation Ub(t).

In fact, the accuracy of the numerical predictions is very sensitive to the adopted τ-γ relationsip, a fact that most users either are not aware of or …. choose to overlook.

““VERY SIMPLEVERY SIMPLE””

HYSTERETIC MODELSHYSTERETIC MODELS

G1

τm

τ*

γ*

τ

γ

(α) Εlastic – perfectly plastic

loading: -τm ≤ τ = γ G ≤ τm

Unloading from (τc,γc): -τ*m ≤ τ* = γ* G ≤ τ*m

where τ*=τ-τc, γ*=γ-γc και τm*=τm+|τc|

-τm

γc

τc


Loop for τ≤τmτu

τ

γ

G/Go=1.0ξ=0

Loop for τ≥τm

τ

γ

Gο1

γ

1G

γο

τ

γ

Gο1

γ

1G

γο

( )

65.0~0oG

G1

2

12

21

4

4

1

4

1,

G

G

G

Go

m

omo

o

o

mo

m

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−===

⎪⎪⎭

⎪⎪⎬

⎫

=

=

πξ

γγ

πγτγγτ

πΕΔΕ

πξ

γγ

γτγτ

ελ

OCR = 1-15 0

3050

100PI=200

1.0

0.8

0.6

0.4

0.2

0.0

10-4 10-3 10-2 10-1 100 10

GGmax

γc%

15

OCR = 1-8

PI = 0

15

30

50

100

200

25

20

15

10

5

0

10-4 10-3 10-2 10-1 100 10

γc%

ξ%

!

Comparison with experimental data

RAPID degradation of shear modulus G . . .

and, even worst, RAPID & EXTREME increase of the

critical damping ratio ξ

What are the consequences of What are the consequences of these differences for the these differences for the prediction of the seismic ground prediction of the seismic ground response?response? γ (%)

ξ(%

)

γ (%)


(a) |τ| ≤ |το|

1Go

1G

τo

τ

τc

γoγc γ

G/Go=1ξ=0

((ββ) ) BiBi--linear linear elastoplasticelastoplastic

c

cc

loadingG

unloading

G reloading

ο

ο

τγ

τ τγ γ

=

−−− =

τ

γc γ

(b) |τ| ≥ |το|

1 1

: ooloading

G Gο

οτ τ τ τ

γ γ γ γ− −

− = ⇒ = +

*

*

1*

, :

2

c c

c

oc

o

unloading from

G G GG

G Gο

οο ο

ο ο

γ τ

τ τ ττ γ γ γτγ γ γ γ γ

τ γγτ τ

⎫= −⎪ = ⎫ −

= − = ⇒ = +⎬ ⎬= ⎭⎪= ⎭

1 1

1

o

o o o

o

GG G GG

G G GGG

ο

ο ο ο

γ γγ γ γ γ γ γ

−= + ⇒ − + =

1 11o

o o o

G GGG G G

γγ⎛ ⎞= − +⎜ ⎟⎝ ⎠

τc


τ

τ

γ γ

β

α

γο

2(γ-γο)

2το

ελπξ

ΕΔΕ

=4

1

( ) ( ) ( ) ( )

( ) ( )

( )( ) 11

2 2 22cos cos cos sin sin

cos cos cos cos

4 1 tan tan

tan4 1

1tan tan 90 ' cot ' o

o

GGG

G

ο ο οο

ο ο

ο ο

γ γ γ γ ττα β α β α β

α β α β

γ γ τ α β

α γ γ τβ β β

− −ΔΕ = + = −

⎫= − −⎪ ⎛ ⎞= ⎪ ΔΕ = − −⎬ ⎜ ⎟

⎝ ⎠⎪= − = = ⎪⎭

ΔΕ

Εελ

( )

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

=

o

o

o

G

G

G

G

GG

E

1

1

112

21

14

4

1

2

1

γγ

γγ

π

τγ

τγγ

πξ

τγ

οο

οο

( ) 14 1o

G

Gο ογ γ τ⎛ ⎞

ΔΕ = − −⎜ ⎟⎝ ⎠

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=

o

o

o

o

o

GG

GG

GG

11

1

1

112

γγγγ

γγ

πξ

ο

G1/G0= 0.25

0.1 1 10 100

γ/γο

0

0.05

0.1

0.15

0.2

0.25

ξ (%

)

γ/γο

ξ(%

)


Comparison with experimental data

γ (%)

What are the consequences of the observed deviations for the prediction of seismic ground response?

““SIMPLESIMPLE””

HYSTERETIC MODELSHYSTERETIC MODELS

(α) The “hyperbolic” model

τm

τm

γ

τ Go1

1

1o

m

G

τγττ

=−

monotonicloading - unloading

1

o

o

m

Gή

Gγ

τγτ

= ⇒+

( ) 1

1 o mo

G GG τ γ−

⎡ ⎤= +⎣ ⎦

G1

The above relation for the G/Go ratio is more or less valid regardless of the unloading-reloading scheme which will be chosen to simulate cyclic loading (see the following paragraphs)GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.44

γ*=γ-γc τ*=τ-τc

Go*=Go τm*=2τm

( )

cm

o

coc

m

co

cc

G

Gή

Gά

γγτ

γγττ

τττ

ττγγρα

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=−

−−

−=−

21

21

1

Cycllic unloading – reloading according to MASING (1926)

(it is popular as it always leads to closed and symmetric loops)

τc

γ

ττ*

γ*

γc

-τm

τm

τ1*-γ1*

τ*

γ*

( )( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

+−−=

=⎥⎦

⎤⎢⎣

⎡−

−−

=

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−=−=

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

==

cmm

mcmm

o

m

m

m

m

o

m

c

m

o

m

o

m

co

m

co

m

G

G

Gd

dE

G

GGG

τττττττττττ

τττ

ττττ

ττ

ττ

τγγτ

ττ

τγ

ττ

τ

ττ

ττγ

*

*22*

**

*

*

**1

*2*

*

**1

****1

2

222

22

2

2

2

1

1

21

1

21

1

122

1

τ2*-γ2*

τ

γ

dE


( )( )

( )( )

( ) ( )

**

* *

* *2*

*0

22

2

2

2

22

2

1 1 1

2 2 21

1 2 41 2 ln

4 Ε

21 2 ln

1 111

c

cm

o m c m

cm

o m c m

ccc c

co

m

o

o

m

o o

d

d G

dG

GG

G

G

G

G

Gwith a

G Ga

τ

τ ττ ττ τ τ τ τ

τ τ τττ

τ τ τ τ

τττ γττ

ξ α α απ π π

ξ απ

τκαι

γ

−Ε= ⇒

− −

−ΔΕ =

− −

Ε = = =⎛ ⎞

−⎜ ⎟⎝ ⎠

ΔΕ= = + + + =

⎡ ⎤= −⎢ ⎥

⎣ ⎦

= =+

∫

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+=

=

−=

−=

m

c

mcm

*

m

o

*

o

c

*

c

*

1

GG

ττ

ττττ

τττγγγ

Cyclic unloading – reloading according to PYKE (1979)

(it may be simpler, but does not provide closed and symmetric hysteresis loops)

τc

γ

ττ*

γ*

γc

-τm

τm


Representative hysteresis loops for the hyperbolic model with unloading-relaoding according to Masing and according to Pyke

HYPERBOLICMODEL & MASING

HYPERBOLICMODEL &

PYKE

⎟⎟⎠

⎞⎜⎜⎝

⎛+=+=

=

−=

−=

m

cmcmm

oo

c

c

GG

ττ

ττττ

τττ

γγγ

1*

*

*

*

( )

ccm

o

coc

m

cm

co

cc

G

Gή

Gά

γγττ

γγττ

ττ

τ

ττττγγρα

−⎟⎟⎠

⎞⎜⎜⎝

⎛

++

−=−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−

−=−

1

1

1

1

τc

γ

ττ*

γ*

γc

-τm

τm

Cyclic unloading – reloading according to PYKE (1979)


similarly . . .

( )

24

42

1

11

1

11

1

21ln

21ln22

2

4

2

1

24

2

22

1

2

22

1

2121

1

1

2

2

+++

++

−=

++=

=

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−++

+−

+=

+++

=

aa

a

aC

aC

Ga

aCaC

aCaCCC

aC

C

aa

aa

G

G

o

m

o

γτ

παξ

Comparison with experimental data …………

What are the consequencesof the observed deviations for the prediction of seismic ground response?


Monotonic loading

1

max 1

max 1

11 1

0.641 0.5625

2

w

y

y

G a

forGw

ττγτ

α ττγτ

−⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= + −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦= ⎫ ⎛ ⎞

= +⎬ ⎜ ⎟= ⎝ ⎠⎭

τc

γ

ττ*

γ*

Unloading-reloading

1*1

*

*

2ττ

τττ

γγγ

=

−=

−=

c

c

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

+−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

= −

max

1

1max

11

12

11

1

1

G

G

w

w

a

G

Gw

c

y

πξ

ττ

y

cmax

1

max

a 0.64 G 1

Gw 21 0.56

2 G1 G3

ττ

ξπ

= ⎫=⎬

= ⎭ +

⎛ ⎞= −⎜ ⎟⎝ ⎠

(b) Ramberg-Osgood (1943)

Representative hysteresis loop for Ramberg-Osgood model


Comparison with experimental data …………

Fair agreement with the experimental data is possible, both for G/Go and ξ-γ, following a proper selection of the model parameters. This is not possible with any of the models presented earlier.

Final Comments on Numerical Methods

The nonlinear analysis with time integration is free from any basic approximations and, , in theory at least, it may provide higher accuracy. However, in practice, it is also subjected to some important limitations which need to be accounted for during application.

First, we must make sure that the code that we are using is equipped with the proper constitutive model for the simulation of cyclic soil response. In addition, extra caution is required for assigning the correct critical damping ratio ξo (or Do) at very small shear strain amplitudes, since this parameter is frequency dependent (Rayleighdamping) and may obtain erroneously high values if we are not careful (e.g. see figure of next page).

In any case, we have to admit that this methodology is the most reliable for applications where intense soil non-linearity is anticipated (e.g. very strong seismic excitations and/or very soft soils).

The basic advantage of the equivalent-linear analysis in the frequency domain is simplicity. On the other hand, this method violates one basic law of Mechanics: it applies supperposition of the harmonic components of ground response despite that soilresponse during seismic loading is non-linear. As a result, the contribution of high-frequency components is underestimated while the contribution of the low frequency components is over estimated. Nevertheless, these effects are not significant, and may be readily overlooked, as long as the shear strain amplitude in the ground does not exceed about 0.03-0.10%.


C = α Μ + β Κ

Example of Rayleighl damping ratio variation with excitation frequency


Homework 4.2Homework 4.2

Fit the elastic – perfectly plastic model to the experimental curves for G/Gmax of Vuccetic & Dobry for G/Gmax=0.50, for the different values of plasticity index ΡΙ. In the sequel, compare the theoretical and the experimental G/Gmax – γ and ξ-γrelations.

What is the maximum ξ value predicted with the various theoretical models presented here? How does this value compare to the maximum experimental values? How important are the observed differences for the prediction of seismic ground response?

Experimental curves for the G/Go-γ and ξ-γ relations (Vucetic & Dobry, 1991)

OCR = 1-8

PI = 0

15

30

50

100

200

25

20

15

10

5

0

10-4 10-3 10-2 10-1 100 10

γc%

ξ%

OCR = 1-15 015

3050

100PI=200

1.0

0.8

0.6

0.4

0.2

0.0

10-4 10-3 10-2 10-1 100 10

GGmax

γc%

SAN

D(I

p =0) SAN

D(I p

=0)

γ (%) γ (%)

ξ(%

)

G /G m

ax

Effect of soil type (through IP ή PI)GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.52

Homework 4.3: Soil effects in Lefkada, Greece (2003) earthquakeHomework 4.3Homework 4.3: : Soil effects in Soil effects in LefkadaLefkada, Greece, Greece (2003)(2003) earthquakeearthquakeThe accompanying figures provide the basic data with regard to the recent (2003) strong motion recording in the island of Lefkada:

-Acceleration time histories and elastic response spectra (5% structural damping) from the two horizontal seismic motion recordings on the ground surface.

-Acceleration time histories and elastic response spectra (5% structural damping) for the two horizontal seismic motion recordings on the surface of the outcropping bedrock, as computed with a non-linear numerical analysis

-Soil profile at the recording site.

(a) Using the equivalent linear method of analysis, COMPUTECOMPUTE the peak seismic acceleration and the elastic response spectra at the free ground surface, using as input the seismic recordings at the outcropping bedrock.

Compare with the actual recordings and comment on causes of any observed differences.

(b) Repeat your computations assuming that soil response is elastoplastic (see Hwk 4.2) and compare with the predictions of (a) above. Comment on the observed differences.

(NOTE: Choose the LONG component of seismic motion for your computations)

-0.4

-0.2

0

0.2

0.4

a(g

)

0 5 10 15 20 25

Time (s)

-0.4

-0.2

0

0.2

0.4

a(g

)

0,34

0 5 10 15 20 25

Time (s)

0,42 Surface

Out.Bedrock

0,200,35

Surface

Out.Bedrock

LONG TRANS


0

0.6

1.2

1.8

Sa(

g)

0 0.4 0.8 1.2 1.6 2

Tstr (s)

0.5

1

1.5

2

2.5

AS

a

0 0.4 0.8 1.2 1.6 2

Tstr (s)

TRANSLONG

ΕδαφικήΕδαφική ΤομήΤομή στηνστην ΘέσηΘέση καταγραφήςκαταγραφής

0 100 200 300 400 500

VS (m/sec)

30.0

CL

MARL(CL)

ML

4.0

12.0

Vb = 450m/s

SC 8.0

6.0

0 10 20 30 40 50

NSPT

30

25

20

15

10

5

0

De

pth

(m

)

>50

>50

>50

>50

>50

>50

0

0.2

0.4

0.6

0.8

1

G/G

max

γ (%)

0

10

20

30

ξ (%

)

H = 24m

Seismicbedrock

12

3

12

3

CH

10-4 10-3 10-2 10-1 1 10

6.0

2.0


4.seismic ground response or soil “amplification” of...

Documents