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

4.Seismic Ground Response4.Seismic Ground Response

orsoil “amplification” of seismic p

ground motions

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

Oct. 2016.

GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.1

4.1 EXAMPLES FROM (REAL) CASE HISTORIES

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:g

Α. Soil “amplification” coefficientpΒ. Important soil and seismic motion parametersC. Frequency dependence of soil amplification effectsq y p p

San Francisco 1957 . . . .

Variation of peak ground acceleration and elastic response spectrain connection with local soil conditions at recording stations . . .


San Francisco 1989 . . . .

Recording atRecording atoutcropping ROCK75 cm/sec2

Recording at the surface Recording at the surface of earth fill on Bay mud

1 0 / 2170 cm/sec2

Soil “amplification” or “de-amplification” ?


Is this a pure soil effect?

Kobe, Japan 1995

stiff soil, amax=0.82g

l fillloose fill, amax=0.32g


Αthens, Greece 1999 . . . .

Chalandri: soil a =0 19gChalandri: soil, amax=0.19g

Demokritus: rock, ,amax=0.06g

Αthens, Greece 1999 . . . .

STIFF SOIL AMPLIFICATION at Ano Liosiaat Ano Liosia municipality

Slight or no damage

Severe damageor collapse

Moderate damage


Cross-section N-S

75

100

ildin

gs N S

25

50

(%)

of

bu

i

0(

0360 7600 1300 0 1300 0 1300 0 1300 0 1300

DH540

20

H (

m)

Vs (m/sec)

BoreholeCHT2

DH5

60

40

DE

PT

_ _ Vs,30 =613

_ Vs,30 =545 _

Vs 30 =500

_ Vs,30 =496

80

Vs,30 =697 m/sec, Vs,30 500

Conglomerates Clayey Marl

Mexico City 1985

thi kn f il Typical recordings on geology thickness of soil

deposits

yp gthe surface of SOIL & ROCK


F

Simplified mechanism of SOIL effects

Fouriercomposition Fourier Spectrum

time

Transfer

time

function

Fourier Spectrum

time

Fourierde-composition frequency

Initial Conclusions :

Α S il “ lifi ti ” ffi i t (0 40 3 20) 0 6 2 0Α. Soil “amplification” coefficient: (0.40-3.20) 0.6-2.0

Β. Important soil and seismic motion parameters:soil & excitation characteristics

C. Frequency dependence of soil amplification effects:amax & Sa

Quite often,

the effect of a few (tens of) meters of soft soilis larger and more significant thanis 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 από μηχανική άποψη μπορούν να εξομοιωθούν με κοκκώδη. Στρώσεις κοκκώδους υλικού μέσης πυκνότητας (και) πάχους μεγαλύτερου τωνκοκκώδους υλικού μέσης πυκνότητας (και) πάχους μεγαλύτερου των 5μ, ή μεγάλης πυκνότητας (και) πάχους μεγαλύτερου των 70μ.

SS TT TT

Seismic Codes: EC-8 ………

DescriptionDescriptionSoil Soil

ParametersParameters

SS TTBB TTCC

M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5

AA 1.0 1.0 0.15 0.05 0.4 0.25

AA BB 1.2 1.35 0.15 0.05 0.5 0.25

CC 1.15 1.5 0.20 0.10 0.6 0.25

DD 1 35 1 8 0 20 0 10 0 8 0 30BB

Deep deposits of

DD 1.35 1.8 0.20 0.10 0.8 0.30

EE 1.4 1.6 0.15 0.05 0.5 0.25

3

CC

p pdense or medium-dense sand, gravel or stiff clay with thickness from

Vs,30=180-360m/s

NSPT=15-50

M>5.5

several tens to many hundreds of m

Cu=70-250KPa 2

e / (S

*ag)

A

B,EC

D

DD

EE

1

Se

EE0 0.5 1 1.5

T (s)

0


SS TT TT

Seismic Codes: EC-8 ………

DescriptionDescriptionSoil Soil

ParametersParameters

SS TTBB TTCC

M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5

AA 1.0 1.0 0.15 0.05 0.4 0.25

AA BB 1.2 1.35 0.15 0.05 0.5 0.25

CC 1.15 1.5 0.20 0.10 0.6 0.25

DD 1 35 1 8 0 20 0 10 0 8 0 30

3

BB

Deep deposits of

DD 1.35 1.8 0.20 0.10 0.8 0.30

EE 1.4 1.6 0.15 0.05 0.5 0.25

CC

p pdense or medium-dense sand, gravel or stiff clay with thickness from

Vs,30=180-360m/s

NSPT=15-50

M<5.5

2

e /

(S*a

g)several tens to many

hundreds of mCu=70-250KPa

1S

A,B,C,E

DDD

EE0 0.5 1 1.5

T (s)

0

EE


4.Seismic Ground Response4.Seismic Ground Response

orsoil “amplification” of seismic p

ground motions


(PART B)

Oct. 2016.


4.2 DEFINITIONS4.2 DEFINITIONS

Soil effects are defined as the ratio of a seismic motion parameter (e.g. peak or spectral acceleration) at the free ground surface divided by the same parameter at the surface ground surface divided by the same parameter at the surface of the bedrock.

i e motion at Γi.e. motion at Γ

motion at Β

or (more often) κίνηση στο σημείο Γ

κίνηση στο σημείο Α

How is “seismic bedrock” defined ?There is no general consensus on this matter. Conventionally, any geological formation with shear wave propagation velocity any geological formation with shear wave propagation velocity Vs≥750m/s may be characterized as seismic bedrock. However, such a unique definition is not always correct, as it is the contrast between the soil and bedrock shear wave velocities that is important. From another point of view, seismic bedrock may be defined as the outcropping geological formation bedrock may be defined as the outcropping geological formation where the seismic excitation was recorded.

For the Greek practice where seismological predictions for For the Greek practice, where seismological predictions for “rock sites” are based on seismic recordings on the free surface of very stiff soils and rocks (i.e. soil category A of EAK 2002) h i i b d k b d fi d i hi N 2002), the seismic bedrock may be defined within Neogene or older geological formations, as well as on well cemented conglomerates resting upon them. conglomerates resting upon them.


In principle, analysis-prediction of soil effects is ibl ith f th th f ll i th d possible with one of the three following methods

(with greatly different complexity, as well as, )accuracy):

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

II Empirical (based on geological and urac

y

icity

II. Empirical (based on geological and seismological input)ac

cu

simpli

III. Numerical (SHAKE, DESRA, …)

s


4.3 ANALYTICAL METHODS4.3 ANALYTICAL METHODS

f l l d b d kUniform elastic soil on rigid bedrock

Uniform visco-elastic soil on rigid Uniform visco elastic soil on rigid bedrock

U f l l fl bl Uniform visco-elastic soil on flexible bedrock

Non-uniform visco-elastic soil on flexible bedrockflexible bedrock

Elastic soil no RIGID bedrock

V ρ ξ HVb, ρb, ξb Vs, ρs, ξs Hb, ρb, ξb

(ή Vr, ρr, ξr)

GDefinitions : GV

ρ

4H 4H

s b rs b

4H 4HT ,T ήT

V V

ωk wave number

V


kzti

kzti

Be

Ae

z H

z 0

i ωt kz i ωt kz

Boundary condition : τ 0

uG 0 Gik Ae Be 0

tieuu

Be

z 0

iωt

G 0 Gik Ae Be 0z

for z 0 : Gik A B e 0 A B

oeuu

ikz ikziωt iωte e

thus : u 2Ae 2Ae cos kz2

kztikzti BeAeu iωt

z H oBoundary condition : u u e

oo

uthus : u 2 Acos kH ή A

2 cos kH

stationary wave with amplitude

kH

kzukzAu oo cos

coscos2

Amplification factor:

111

ίF

factor:

2

coscos

.

1

TTkHί

exc

soil

,2,1,12 nnTT

exc

soil Resonance:

12cos

zn

u Modal shapes:

1cos

122

cos2

nz

Hn

A

Modal shapes:

2,2

3cos

1,2

cos

2n

H

z

nH

A

uή

3,2

5cos

,22

nH

zHA

ή

2 H


|F1|

0 1 3 5 7 Ts/Texc

00 1-1 u/2A

0

z/H

1

1

1 1

cos

motionF

motion kH T

Amplification f t

.

coscos

2soil

exc

motion kH T

T

T

factor:

|F |2 1, 1,2,soil

exc

T n nT συντονισμός:|F1|

cos 2 12 2

u zn

A H

ιδιομορφές: 2 2

cos , 1

A H

zn

ιδιομορφές:0 1 3 5 7 Ts/Texc

,2

3cos , 2

2 2

Hu z

ή nA H

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

2 25

cos , 32

ήA H

zn

H

(motion Α) = (motion Β)And consequently

motion motionF ( )

2 H1F ( )ί motion


Visco-elastic soil on RIGID bedrock

Vs, ρs, ξs HVb, ρb, ξb(ή Vr, ρr, ξr)

Definitions for elastic Definitions for visco-elastic soil (ξ=0)

G

soil (ξ≠0)

G G i * 1 2 G

Vρ

4H 4HT T ή T

G

V V i *

* 1 0,30

s b rs b

T ,T ή TV V

ωk wave number

H HT i T i

V V *

*

*

4 41 1

k wave numberV k i k i

V V *

* 1 1

According to the “correspondence principle”, you may use the solution for elastic soil, with the real soil variables replaced b th di l i bl Th by the corresponding complex variables. Thus,

kHikHikHHkF

cos

1

1cos

1

cos

1*2

Amplification factor

0cossinhcoscos 2222 yyxyxiyxor, given that:

2 2

1/ sF where V

g

2 22coss

kH kH

2 1F 2max

2 1F

n

when

2 1 2 12

soil

exc

TkH n or n

T


Elastic soil on RIGID b d kbedrock

Visco-elastic l soil on

RIGID bedrockbedrock

Uniform visco-elastic soil on FLEXIBLE bedrockbedrock

Boundary conditions:

tissss

sss ezkAu

uGz

cos20:0 *

Boundary conditions:

tiezkAu *cos2

b

bb

s

ssbssb

sssss

ss

z

uG

z

uGuHuHzz

z

&0:,0

zs sss ezkAu

scos2

Hbs

HikHik

sbss eaeaAA** ** 11

2

1ρs,Vs

ρb,Vb

HikHiksb

sb

ss eaeaAB** ** 11

2

12zb bb zkti

beA*

ssssss ia

iVVa

11**

bb zktibeB

*

bbbbbb ia

iVVa

11*


BA

A2

0u

0u

u

uF

bb

s

b

s

B

A3

)(F

HkHkcos

1

Hkcos

1F 2

21

22*

s

3

bbbB

HkHkcos ssss

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

UA/UΓ:

sA A2uF

b

4

11F

A2uF

21

2

sss

2*

s

*

s

*

s

4

HkHkcosHksiniaHkcosF

για άκαμπτο βράχο Vb ∞ a 0 ξ ξ .excTa

2

ό

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

F1

F ss

s

sss

Va

Ta

32

12

sss

24 F

HkHkcosF

bb

s Va


Important soil and excitation parameters affecting soil amplification

11

soil amplification . . .

21

2

sss

2*

s

*

s

*

s

4

HkHkcos

1

HksiniaHkcos

1F

.2: ,exc s s

s s s ss b b

T Vwhere a a

T V

Problem parameters:

1

exc

s

s

s T

T

2V

HHk

TT

s

excs

1

exc

ssss T

Ta

2

)T

Tή(

T

T

V

Va

exc

b

S

b

bb

sss


4.4 EMPIRICAL METHODS4.4 EMPIRICAL METHODS

Analysis of damage from previous earthquakes

C l ti t ( f ) lCorrelation to (surface) geology

Attenuation relationships for αmax, Vmax, ….u p f max, Vmax, .

Correlation to shear wave velocity VS

Seismic Codes

S mi l ti l l ti hiSemi-analytical relationships

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


Recent Seismic Codes: π.χ. EC - 83

M>5.53

M<5.5

2

e / (

S*a

g)

A

B,EC

D2

e /

(S*a

g)

1

Se

1

Se

A,B,C,E

D

0 0.5 1 1.5T (s)

00 0.5 1 1.5

T (s)

0

SS TTBB TTCC

M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5 M>5,5M>5,5 M<5,5M<5,5

AA 1 0 1 0 0 15 0 05 0 4 0 25AA 1.0 1.0 0.15 0.05 0.4 0.25

BB 1.2 1.35 0.15 0.05 0.5 0.25

CC 1.15 1.5 0.20 0.10 0.6 0.25

DD 1.35 1.8 0.20 0.10 0.8 0.30

EE 1.4 1.6 0.15 0.05 0.5 0.25

Parameters Ground

type Description of stratigraphic profile

Vs,30 (m/s) NSPT (bl /30 )

cu (kPa) , ( )(blows/30cm)

( )

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

800 _ _

Deposits of very dense sand gravel or

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

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

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

A soil profile consisting of a surface

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

D i f li fi bl il f i iS2

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


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

700

800Vb=800m/s

700

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

500

600

(m/s

ec)

500

600

(m/s

ec)

(m/s

)

B

300

400

Vs

,30

(

B

300

400

Vs

,30

(

Vs,

30(

A

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

A

C

DE0 10 20 30 40 50 60 70 80

H (m)

100

200

H (m) H (m)

1.6

1.7

1.8EC8

Vb=800m/s

V =1200m/s

H (m) H (m)H (m) H (m)

1.2

1.3

1.4

1.5

S

Vb=1200m/s

Average

0.9

1

1.1

A B C D E

(M > 5.5)

700

800Vb=800m/s

700

800Vb=1200m/s

500

600

700

sec)

500

600

700

sec)

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

B

300

400

Vs,

el (

m/s

B

300

400

Vs,

el (

m/s

(m/s

)

A

C

DE0 10 20 30 40 50 60 70 80

100

200

A

C

DE0 10 20 30 40 50 60 70 80

100

200VS

H (m) H (m)

H (m) H (m)

1 5

1.6

1.7

1.8EC8

Vb=800m/s

Vb=1200m/s

( ) H (m)

1.2

1.3

1.4

1.5

S

Vb 1200m/s

Average

0.9

1

1.1

A B C D E

(M > 5.5)GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.23

800Vb=800m/s

800Vb=1200m/s

600

700

Vb=800m/s

600

700

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

A1 A2 B400

500

Vs,

el (

m/s

ec)

A1 A2 B400

500

Vs,

el (

m/s

ec)

(m/s

)

C200

300

V

C200

300

V

VS

(

A2 DE0 10 20 30 40 50 60 70 80

H (m)

100A2 DE0 10 20 30 40 50 60 70 80

H (m)

100

H (m) H (m)

1 6

1.7

1.8EC8

Suggested

H (m) H (m)

1 2

1.3

1.4

1.5

1.6

S

Vb=800m/s

Vb=1200m/s

Average

0.9

1

1.1

1.2

A1 B C D EA2

(M > 5.5)


Semi-analytical relationships (Bouckovalas & Papadimitriou, 2003)(Bouckovalas & Papadimitriou, 003)

REAL conditionsIDEALIZED conditions

A CC A

linear visco-elastic soil nonlinear visco-elastic soil

ρs , ξs , Vs T 4H / V

layer i: ρsi , ξsi , VsiTs= 4H / Vs H

BB´

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

TeT = Te

Semi-analytical relationships (Bouckovalas & Papadimitriou, 2003)

REAL conditions

(Bouckovalas & Papadimitriou, 003)

IDEALIZED conditions

A CC A

linear visco-elastic soil nonlinear visco-elastic soil

ρs , ξs , Vs T 4H / V

layer i: ρsi , ξsi , VsiTs= 4H / Vs H

BB´

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

Soil ‘Amplification’ =motion at A

Soil ‘Amplification’ =motion at C


Database with more than 700 SHAKE-type analyses

Sit Ch t i ti E it ti Ch t i tiSite Characteristics Excitation Characteristics

Ts = 0.04–3.33 s

120

0.04

-0.1

0.2-

0.3

04-

0.5 0.6-

0.8

1.0-

1.50

H = 3.5–240 m

120

10-1

5 20-2

530

-35

45-5

5

00

abmax = 0.01–0.45g

300

07 1 4

0,41

-0,4

5

0

60

0 0.4 1

0

60 0-5 1 3

65-1

015

0-24

0

0

100

200

0,01

-0,0

0,15

-0,2

1

0,26

-0,3

4

00 50-2

00

300

00

Vs = 50–700 m/sVb = 100–1000 m/s

400 50

n = 0.5–24200 0.

51

0

60

120

<100

150

250-

3

350-

400

450-

600

0

200

400

103

408

55

600 65

0

750

>800

0

100

200

1.5

2.5

3.5 4

4.5 5 6

724

Step 1: Non-linear Soil Period, Tsp s

3)

on

best fit relation:1

2

ctio

ns

over

estim

ation

by 5

0%

best fit relation:

(pre

dic ov

perfectagreement 041b

0.1sec

(

stim

ation

3%

g 31

041

53301.

.max

,os

b

ossV

aTT

TS :

s

unde

rest

by 3

3%

0 03

,osV

0.1 1 2 3

TS : sec (data)0.03


Step 2: ‘Amplification’ of peak d l i Aground acceleration, Aa

Data

1 5

2Data

Tb/TS = 0.05 - 0.40 n = 0.5 - 2

abmax = 0.41 - 0.45

1

1.5

Aa

0.5 best fitrelation

b t fit l ti

0 1 2 3 4 5 6

Ts / Te

0

best fit relation:

211 es /TTCA

n

naC b

121

1701

.max.

T

222

221 eses

a

/TTC/TT

A

s

b

T

TC 5700512 ..

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

0.4

g)

2excitation 1

-0.4

0

a (

g

0 41.5

1.75

a

0 4

0

0.4

a (

g)

1

1.25

AS

a / A

aexcitation 2fit relation

ASa,p*

0 4 8 12

time (sec)

-0.4

0 9

1.2 0.75

1

AS

a* =

A

ASa,r*fit relation:

0.3

0.6

0.9

Sa (

g)

0.25

0.5

A

commonab = 0 3g

2222

21

1

1 sstrSa

/TTB/TT

/TTBA

*

0 0.4 0.8 1.2 1.6 2

Tstr (sec)

00.01 0.1 1 10

Tstr / TS

0

a max 0.3g

*A1

21 sstrsstr /TTB/TT

*,rSaAB 1

*

*

,

,

pSa

rSa

A

AB

12


Peak Spectral ‘Amplification’, ASa,p*8

Datan= 0.5 - 1

4

6

A *

Tb/TS= 0.05-0.3ab

max=0.40-0.45g

2

4ASa,p

best fit

best fit relation:

0 1 2 3 4 5 6 7 80

2 best fitrelation

relation:

ss

T

T

T

T131801

0580

,..

0 1 2 3 4 5 6 7 8

TS / Te

e

s

e

s

ee

pSa T

T

T

TD

TT

A 4113181 ,.*,6130

5040

2790 ..

.

n

T

TD

s

b

e

s

T

TD 433181 ,.

Residual Spectral ‘Amplification’, ASa,r*

6

8

Datan=0.5 - 1.5

Tb/TS= 0.05-0.4

4

6

ASa r*

Tb/TS 0.05 0.4ab

max= 0.01-0.45g

2

Sa,r

best fit

0 1 2 3 4 5 6 7 8 9 100

best fitrelationbest fit

relation:

Ts / Te

e

s

e

s

T

T

T

T130201 ,.

e

s

e

s

ee

rSa T

T

T

TF

TT

A 6116980 ,.*,4060

4740

1890 ..

.

n

T

TF

s

b

e

s

T

TF 656980 ,.


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

San Gabriel

PacoimaReservoir

San GabrielMts

LDF

H

Santa Susana Mts LDF

RRS

HWY 180HansenDam

HWY 170

SFY

SAN FERNANDO VALLEY

I-5I-405

HWY 101

HWY 170 VerdugoMtsN Northridge

Epicenter

HWY 101

HWY 101

Santa Monica MtsSanta Monica Mts

0 1 2 3 miles

Soil conditions at recording sitesgLDF 0 300 600 900

VS,o (m/s)

00RRS

SiltSFY

Clay -Silt

1010

Gravel

Silt

sand -gravel

LDF

LDF: ‘bedrock’ site

Silt

30

20

30

20

m)

FineSand

Silt

RRSSand -

4040

epth

(m

Silt-SandStone

Silt

RRSSFY

‘soil’ sitesSand Gravel

60

50

60

50de

Gravel

SFYRRS

DenseSand

70

60

70

60 SFYRRS

(Gibbs et al. 1999, Cultrera et al. 1999)8080

DenseSand

very hardsilt stone


Site response at SFY (Arleta Fire Station)

Approx. Relations Num. Predictions Approx. Relations

2 2 2vs Records vs Records vs Numer. Predictions

1oil

(g)

1 1

Saso

0.01 0.1 1

T t (sec)

00.01 0.1 1

T t (sec)

00.01 0.1 1

Tstr (sec)

0

Tstr (sec) Tstr (sec) str ( )

Site response at RRS (Rinaldi Receiving Station)

Approx. Relations vs Records

Num. Predictions vs Records

Approx. Relationsvs Numer. Predictions

2

g)

2 2vs Records vs Records vs Numer. Predictions

1

Saso

il (g

1 1

0 01 0 1 10

S

0 01 0 1 10

0 01 0 1 10

0.01 0.1 1

Tstr (sec)0.01 0.1 1

Tstr (sec)0.01 0.1 1

Tstr (sec)

Approx. Relations Numerical Predictionspp


Homework 4.1Homework 4.1: : Soil effects in Lefkada, GreeceSoil effects in Lefkada, 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:( ) g g

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

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

-Soil profile at the recording site.

U i th i l ti l l ti hi COMPUTECOMPUTE th d f t l 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)(NOTE: Choose the LONG component of seismic motion for your computations)

0 2

0.4 0,34 0,42 SurfaceSurface

LONG TRANS

0 2

0

0.2

a(g

)

-0.4

-0.2

0.2

0.4

)

Out.BedrockOut.Bedrock

-0.2

0

a(g

0,20

0 5 10 15 20 25

Time (s)

-0.4

0 5 10 15 20 25

Time (s)

0,35


TRANSLONG

1 2

1.8TRANSLONG

0.6

1.2

Sa(

g)

0

2.5

2

2.5

a

1

1.5AS

a

0 0.4 0.8 1.2 1.6 2

Tstr (s)

0.50 0.4 0.8 1.2 1.6 2

Tstr (s)

Soil profile at the recording siteSoil profile at the recording siteSoil profile at the recording siteSoil profile at the recording site

V (m/sec)N γ (%)0 100 200 300 400 500

VS (m/sec)

CL

4 0

0 10 20 30 40 50

NSPT

0

0.8

1

γ (%)10-4 10-3 10-2 10-1 1 10

2.0

ML

4.0

SC 8.0

6.0

10

5

0.4

0.6

G/G

max

12

3

CH6.0

MARL

ML12.0

15

10

pth

(m

)

>500

0.2

30

1

MARL(CL)

20

De

p

>50

>5020

30

%)

H 241

2

Vb = 450m/s25 >50

>50

10

ξ (%H = 24m

Seismicbedrock

23

30.030 >50 0


4 Seismic Ground Response4.Seismic Ground Response

orsoil “amplification” of seismic soil amplification of seismic

ground motionsground motions


(PART C)(PART C)

Oct. 2016.


4.4 NUMERICAL METHODS4.4 NUMERICAL METHODS

NON-LINEAR

TIME DOMAIN ANALYSIS

EQUIVALENT LINEAR

FREQUENCY DOMAIN ANALYSIS

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

(1)

Response( )

(i) hiρi, Goi, ξoiat the free ground surfaceor at theinter face between

ρb, Gb (ξb=0)Excitationt th inter-face between

layersat the “outcropping”or at theor at th“buried”ΥΠΟΒΑΘΡΟ


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

NONUNIFORM (layered) visco-elastic soil on flexible

ti t l

ybedrock.

motion at layer m

tizikzik eeBeAu mmmm **

u1

z11 ρ1, V1, ξ1

tizikzikm

mmm

eeBeAkiGu

G

eeBeAu

mmmm

mmmm

*****

um

zmm ρm, Vm, ξm

u mmmmm

mm eeBeAkiGz

Gm+1 ρm+1, Vm+1, ξm+1

um+1

zm+1

i

timmom eBAu

**

, για zm=0:

tihikhik

timmmmom

eeBeAu

eBAkiG

mmmm

**

**,

για zm 0

tihik

mhik

mmmhm

mmhm

eeBeAkiG

eeBeAu

mmmm ****

,

,

για zm=hm:

Boundary constraints at the free ground surface

A0tiezkAu

BA

1*111

110,1

cos2

0

tiezkkiGA 1*1

*1

*111 sin2

B und r c nstr ints b t n l rsBoundary constraints between layersmmmm hik

mhik

mmmhmm eBeABAuu**

1101

mmmm

m

m

hikm

hikm

mmmmhmm

mmmmhmm

eBeAGk

GkBA

**

**

**

11,0,1

11,0,1

m

mm Gk 11,,

and finally

1 1 * ** **1`

*

1 11 1

2 2m m m mik h ik h

m m m m mm m

A A a e B a eV

ό a

* *

** * 1 1

1`

1 11 1

2 2m m m m

mik h ik h m m

m m m m m

ό aV

B A a e B a e


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

i 1 * ** *1 1ik h ik hi=1

1 1 1 1* *2 1 1 1 1

* * *

1 11 1

2 2

cos sin

ik h ik hA A a e A a e

A k h ia k h

or, * *

1 1 1 1

1 1 1 1 1 1

* *2 1 1 1 1

cos sin

1 11 1

2 2ik h ik h

A k h ia k h

B A a e A a e

*2 1 1 1 1A f k h A

briefly

2 1 1 1 1

* * *1 1 1 1 1 1

2 2

cos sinA k h ia k h

2 1 1 1 1

*2 1 1 1 1

f

B g k h A

i=2 2*22

*2 *

22*223 1

11

1 hikhik eaBeaAAi 2

2

*22

*2 *

21*211

22223

11

11

22

hikhik eaeaaA or,

briefly

2*22

*2 *

21*2113

21211

11

11

22

hikhik eaeaaAB

* *3 2 1 1 2 2 1,A f k h k h A

y

212113 1

21

2eaeaaAB

* *3 2 1 1 2 2 1,B g k h k h A

iAm=Α1•fm-1(k1

*h1, k2*h2,…, km

*hm)i=m

Bm=A1•gm-1(k1*h1, k2

*h2,…, km*hm)

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

Γ ΑΓ Α

A 1 13

B m m m 1 m 1

u A B 2F

u A B f g

Β’ Β

B m m m 1 m 1

A A4

B B'

f g

u u uF

u uu

B B'

224 3 b b b

u uu

F F ( ) cos k H k H

(uB ≈ uB’)


The basic input data required for this type of analysis 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

h l h d l G d h f 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 y p g yshear strain amplitude (G/Go-γ and ξ-γ curves)

±τ

±±γc

1

Go1

G < Go (απομείωση)τ

1

∆Ε

Εελ.

γc

-γc

γ

ελ.

1 .4

• shear modulus degradationcyclic

loadingearthquake

• hysteretic energy loss

loadingGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.37

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

PI = 025

1.0

OCR = 1-8

15

3020

PI 200

0.8

0.6G 50

100

200

15

10

ξ%

1530

50100

PI=200

0.4

GGmax

200

5

OCR = 1-15 015

0.2

0

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

γc%

0.0

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

% (%) γ (%)γc%γ (%) γ (%)

Eff f il ( h h ή P )Effect of soil type (through IP ή PI)

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

(Ishibashi 1992) (Ishibashi 1992)

m ,PI mo'm

max

0.492

GK , PI

G

0.000102 n PIK PI 0 5 1 tanh ln

6 1.4040.4

1 3

K , PI 0.5 1 tanh ln

0.0 PI 0

3.37 10 PI 0 PI 150 000556

1.3

o0.000556

m , PI m 0.272 1 tanh ln exp 0.0145PI n PI7.0 10

7 1.976

5 1.115

PI 15 PI 70

2.7 10 PI PI 70


Solution Sequence

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

2nd Step: Gl=Go,l και ξl=ξο

3 d S C i f f f i F f h il 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 g p p pthe seismic ground response

6th Step: Computation of maximum shear strain amplitudep p pγmax at the middepth of each soil layer

7th Step: Computation of the shear modulus G and damping ratio ξ7 Step: Computation of the shear modulus G and damping ratio ξvalues which correspond to 2/3 γmax.


4.4 NUMERICAL METHODS4.4 NUMERICAL METHODS

NON-LINEAR

TIME DOMAIN ANALYSIS

EQUIVALENT LINEAR

FREQUENCY DOMAIN ANALYSIS

NON-LINEAR ANALYSISNON L NE N LYS S(time domain analysis)

response1

ihi ρi, Goi, ξoi

response

i oi oi

excitationThe basic input data include:

Acceleration time history Acceleration time history for the seismic excitation.

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

The depth and the thickness of each soil layer iThe 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 shear stress-strain relationship (τ-γ) for monotonic and li l di f h il l

of each soil layer i

cyclic loading of each soil layerGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.40

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

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

bMX KX MU

M

M

M 2

1

2211

11

0

00

KKKK

KK

Kith

nM

M

2

3

3322

2211

00

0

K

KKKKKwith

n 300 KGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.41

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

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

Given that G=f(τ ή γ), it is evident that also Κ=f(U). Hence we deal with a non linear differential equation 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).

Regardless of the solution algorithm, it is mandatory to d fi th h t t i l ti hi ( ) hi h

p ( g g )

define the shear stress-strain relationship (τ-γ) which controls the dynamic response (loading-unloading-

reloading) of each soil layer.reloading) of each soil layer.

In fact the accuracy of the numerical predictions is very 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 SIMPLE”“VERY SIMPLE”

HY TERETIC MODELHY TERETIC MODELHYSTERETIC MODELSHYSTERETIC MODELS

τ*

(α) Εlastic – perfectly plastic

τm

ττ

G1

γ*τc

1

γγc

loading: -τm ≤ τ = γ G ≤ τm-τm

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

| |where τ*=τ-τc, γ*=γ-γc και τm*=τm+|τc|GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.42

Loop for τ≤τmτu

τ

G/Go=1.0 τ Gτ Gγξ=0

τ1

Gτ 1G

γγγο γγγο

Loop for τ≥τm Gο1

ο

Gο1

ο

G

1

21

4

4

1

4

1,

G

G

G

Goomo

m

m

G2

244GG mo

o

mo

65.0~0oG

G1

2

1.0

0 8

Comparison with experimental data

50100

PI=200

0.8

0.6

0 4

GGmax

experimental data

OCR = 1-15 0

30500.4

0.2

15

0.0

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

γc%γ (%)

PI = 0

15

25 !RAPID degradation of shear modulus G . . .

γ ( )

OCR = 1-8

15

30

50

20

15ξ%

and, even worst, RAPID & EXTREME increase of the (%

)

50

100

200

15

10

ξ%EXTREME increase of the critical damping ratio ξ ξ

5What are the consequences of What are the consequences of these differences for the these differences for the

0

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

γc%

prediction of the seismic ground prediction of the seismic ground response?response? γ (%)


(β) (β) BiBi--linear elastoplasticlinear elastoplastic

| | | |(a) |τ| ≤ |το|

G

τ

Go

1G

τoτc c loading

G

1o

γoγc γc

c

unloading

G reloading

G/G =1

g

G/Go=1ξ=0

τ(b) |τ| ≥ |το|τc

: oloading

γc γ1 1

: oloadingG G

*

, :c cunloading from

*

1

c

oc

o

G G GG

G G

1* 2o

G1 1

1

o

o o o

GG G GG

G G GG

1 o o o

oG

G G 1 11o

o o o

G GGG G G


τ2(γ-γο)

τ α

4

1

β 2το

γ γγο

ΔΕ

Ε

2 2 22

Εελ

2 2 22cos cos cos sin sin

cos cos cos cos

4 1 t t

11

4 1 tan tan

tan4 1

1

GGG

1tan tan 90 ' cot ' o

o

G

G

14 1G

4 1oG

E2

1G1/G0= 0.25

0.25

oGG1

1

141

1/ 0 .

0.2

o GG 111

2

214

0.15

(%)

(%)

o

o

GG111

0.1

ξ (

ξ (

o G111

0.05

o

o

GG

G

11 1

112

0.1 1 10 100

γ/γο

0

γ/γο

oo GG1


Comparison with experimental data

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

γ (%)

“SIMPLE”“SIMPLE”

HY TERETIC MODELHY TERETIC MODELHYSTERETIC MODELSHYSTERETIC MODELS

(α) The “hyperbolic” model

τmτ Go

1monotonicloading - unloading

G1

γ1

τm

γ1o

m

G

m

oGή

G

1

1 o mG GG

1 o

m

ήG

o moG

The above relation for the G/Go ratio is more or less valid re ardless f the unl adin rel adin less valid regardless of the unloading-reloading scheme which will be chosen to simulate cyclic loading (see the following paragraphs)


Cycllic unloading – reloading according to MASING (1926)

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

* *τ

τ*

γ*=γ-γc τ*=τ-τc

Go*=Go τm*=2τmτc

τmτc

γ*

co

cc G

ά

1

1γγc

m

co

G

2

1

co

coc

G

Gή

21

-τm

m 2

***

τ1*-γ1*

c

oc

o GGG

***

*1

122

11 γ1

m

om

o

*

**1

1

2

* *

τ

m

oG

*1

21

τ2*-γ2*

dE

τ*

Gd

dE

*

**1

*2*

11γτ

m

c

m

oGd

*

12

12

1

γ*

m

m

m

m

oG

**

*

22

2

2

2

cmm

mcmm

o

m

G

*

*22*

2

222


** 2d

*

* *

2

2cm

o m c m

d

d G

* *2

**

0

22

2

ccm

o m c m

dG

0

221 1 1

2 2 2

o m c m

ccc c G

2 2 21 c

om

GG

21 2 41 2 ln

4 Ε o

G

G

21 2 ln

o

G

G

1 111

o

m Gwith a

G G

1o oG G

a

Cyclic unloading – reloading according to PYKE (1979)

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

*τ

τ*

c

*

c

τc

τm

o

*

o GGτc

γ*

m

c

mcm

*

m 1

γγc

-τm


Representative hysteresis loops for the hyperbolic model ith l di l di di t M i d di with unloading-relaoding according to Masing and according

to Pyke

HYPERBOLICMODEL &

PYKE

HYPERBOLICHYPERBOLICMODEL & MASING

C li l di l di di PYKE (1979)Cyclic unloading – reloading according to PYKE (1979)

c*

*

ττ*

oo

c

GG

*

τc

τm

m

cmcmm

1*

τcγ*

co

cc G

ά

1

1γγc

m

cm

co

1

1

coc

m

G

Gή

-τm

ccm

oG

1


similarly . . . y

24

22

2

aa

G

G

21ln

21ln22

41

24

22

2121

1

2

aCaCCCC

aaGo

1

22 22

1121

1

a

aCaC

aCaCCC

aC

m

11

C

Ga

o

4211

11

22

1

aC

aC

241 22 aaa

Comparison with experimental data …………

What are the consequencesWhat are the consequencesof the observed deviations for the prediction of seismic

d ?ground response?


(b) Ramberg-Osgood (1943)

Monotonic loading τMonotonic loading

11

1 1

w

τc

ττ*

max 1

1 1

0 64

yG a

c

γ

γ*

max 1

0.641 0.5625

2

yfor

Gw

Unloading-reloading

*

*

c

c

1max

11

1

1

G

Gw

cy

cmax

a 0.64 G 1

Gw 21 0 56

1*1 2

c

1

112

11

Gw

a y

cmax

1

1 0.56

2 G1

max

11 Gw

max

2 G1 G3

Representative hysteresis loop for Ramberg-Osgood model


Comparison with experimental data …………

Fair agreement with the experimental data is possible, both for G/Go and 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 , , y , y p g y , p ,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 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 (Rayleigh small shear strain amplitudes, since this parameter is frequency dependent (Rayleigh damping) and may obtain erroneously high values if we are not careful (e.g. see figure of next page).

I h t d it th t thi th d l i th t li bl f 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 soil pp pp p p g p presponse 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%. GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.52

Example of Rayleighl damping ratio variation ith it ti fwith excitation frequency

C = α Μ + β Κ


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 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 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)y

PI = 025

1.0

OCR = 1-8

15

3020

PI 200

0.8

0.6G 50

100

200

15

10

ξ%

1530

50100

PI=200

0.4

GGmax

200

5

OCR = 1-15 015

0.2

0

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

γc%

0.0

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

% (%) γ (%)γc%γ (%) γ (%)

Eff f il ( h h ή P )Effect of soil type (through IP ή PI)GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 4.54

Homework 4.3Homework 4.3: : Soil effects in Lefkada, GreeceSoil effects in Lefkada, Greece (2003)(2003) earthquakeearthquakeHomework 4.3Homework 4.3: : Soil effects in Lefkada, GreeceSoil effects in Lefkada, Greece (2003)(2003) earthquakeearthquakeqqqqThe accompanying figures provide the basic data with regard to the recent (2003) strong motion recording in the island of Lefkada:g g

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

f surface.

-Acceleration time histories and elastic response spectra (5% structural damping) for the two horizontal seismic motion recordings on the surface of the 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. p g

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

i t th i i di t th t i b d k as input the seismic recordings at the outcropping bedrock.

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

(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 ) p p ( )observed differences.

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

0.4 0,34 0,42 SurfaceSurface

LONG TRANS

0

0.2

a(g

)

-0.4

-0.2

a

0.4 Out.BedrockOut.Bedrock

0

0.2

a(g

)

-0.4

-0.2

a

0,200 35

0 5 10 15 20 25

Time (s)0 5 10 15 20 25

Time (s)

0,35


1.8TRANSLONG

1.2

Sa(

g)

0

0.6

S

0

2.5

1.5

2

AS

a

1

0 0.4 0.8 1.2 1.6 2

Tstr (s)

0.50 0.4 0.8 1.2 1.6 2

Tstr (s)

Εδαφική Τομή στην Θέση καταγραφήςΕδαφική Τομή στην Θέση καταγραφήςΕδαφική Τομή στην Θέση καταγραφήςΕδαφική Τομή στην Θέση καταγραφής

0 100 200 300 400 500

VS (m/sec)

CL

0 10 20 30 40 50

NSPT

0 1

γ (%)10-4 10-3 10-2 10-1 1 10

CL

4.0

SC6.0

50.6

0.8

Gm

ax

3

CH6.0

2.0

ML12.0

SC 8.0

10

m) 0.2

0.4G/G

12

3

MARL(CL)

15

De

pth

(m >50

>50

0

30

( )

20

D

>5020

(%)

H = 24m1

2

30.0

Vb = 450m/s

30

25 >50

>50

>50 0

10

ξ

Seismicbedrock

3

30.030 >50 0


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

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