4.Seismic Ground Response or soil “amplification” of...
-
Upload
nguyenkhanh -
Category
Documents
-
view
218 -
download
2
Transcript of 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” ?
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.3
Soil “amplification” or “de-amplification” ?
Is this a pure soil effect?
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.4
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.5
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.7
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….
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.8
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 ………
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.9
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
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
M<5.5
Seismic Codes: EC-8 ………
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.10
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.11
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.12
( ) ( )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:
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.13
|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
= =
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.14
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.16
( ) ( )( )
( ) ( ) ( ) ( )[ ])(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:
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.17
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 . . .
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.18
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.19
Correlations to (surface) geology:
Factors of average spectral amplificationAtt
enua
tion
Relat
ions
hips
for
α max, V m
ax, ….
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.20
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
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
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.21
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)
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.23
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.24
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 +=
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.25
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
+−
+=*
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.26
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.27
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.28
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.29
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.30
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.31
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.34
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.35
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.36
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.
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.37
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.38
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 . . .
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.39
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.40
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 (%)
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.41
(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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.42
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(%
)
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.43
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.45
( )( )
( )( )
( ) ( )
**
* *
* *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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.46
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)
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.47
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?
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.48
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.49
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%.
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.50
C = α Μ + β Κ
Example of Rayleighl damping ratio variation with excitation frequency
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.51
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.53
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
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 4.54