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8. Seismic Analysis of Slopes
Αντισεισμικός Σχεδιασμός ΠρανώνΕπιχωμάτων
8. Seismic Analysis of Slopes
Αντισεισμικός Σχεδιασμός ΠρανώνΕπιχωμάτων
με την πολύτιμη συμβολή του
Αχιλλέα ΠαπαδημητρίουΛέκτορα στο Πανεπιστήμιο Θεσσαλίας
Πρόσθετο Διάβασμα:Πρόσθετο Διάβασμα:Steven Kramer:Chapter 10 (10.1 έως και 10.6.1)
CONTENTSCONTENTS
8.1 The “pseudo static” approach: Basic Concepts
8.2 Review & Evaluation of seismic coefficients proposed in the literature
8.3 A new Integrated Approach
8.4 Concluding Remarks
H
iθ
W Τ
Ν
Fh
Fv
slidingslope
avWFv =+ = kvWg
cosθFh)sinθFv(Wsinθ] tanφFh)cosθFv[(WcL
FSd+−
−−+=
ahW Fh = = khWg
ah(t)
aV(t)
kh = ah/g
kv = av/g
8.1 The “Pseudo Static” approach: BASIC CONCEPTS
FSd > 1 safe conditions
FSd < 1 slope failure (dynamic) ?
FSd > 1 safe conditions
FSd < 1 slope failure (dynamic) ??
Observe that when the horizontal inertia force Fh increases, then the factor of safety decreases drastically, as the nominator decreases and the denominator increases.
Observe that when the horizontal inertia force Fh increases, then the factor of safety decreases drastically, as the nominator decreases and the denominator increases.
On the other hand, the effect of the vertical inertia force Fv on the nominator and the denominator is similar, so that the overall effect on the factor of safety is much less. For this reason, as well as due to the fact that it is rather unlikely to have the peak Fh and Fv acting simultaneously, we often neglect Fv or consider a reduced value.
On the other hand, the effect of the vertical inertia force Fv on the nominator and the denominator is similar, so that the overall effect on the factor of safety is much less. For this reason, as well as due to the fact that it is rather unlikely to have the peak Fh and Fv acting simultaneously, we often neglect Fv or consider a reduced value.
Dynamic Slope Failure (FSd<1.0): . . . . and so what?
Dynamic Slope Failure (FSd<1.0): . . . . and so what?
When FSd becomes less than 1.0,
the soil mass above the failure surface will slide downslope
as in the case of a “sliding block on an inclined plane”
HOWEVER,
unlike STATIC FAILURE which lasts for ever,
SEISMIC FAILURE lasts only for a very short period (fraction of
a second), as . . . . . .
“Sliding Block” kinematics
(for the simplified case of sinusoidal motion)
base motion
sliding blockmotion
Seismic failure & downslope sliding
Relative Velocity
Relative Sliding
Computation of Relative Sliding ….Computation of Relative Sliding ….
NEWMARK (1965)NEWMARK (1965)
.
.
.
( )CRmax
max CR
aV.
a a
−⎛ ⎞δ = ⋅ ⋅⎜ ⎟⎜ ⎟
⎝ ⎠
2
2
10 50
max
max CR
V.
a a
⎛ ⎞δ ≈ ⋅ ⋅⎜ ⎟⎜ ⎟
⎝ ⎠
2
2
10 50
RICHARDS & ELMS (1979)RICHARDS & ELMS (1979)
max
max CR
V.
a a
⎛ ⎞δ ≈ ⋅ ⋅⎜ ⎟⎜ ⎟
⎝ ⎠
2
4
10 087
E.M.Π. (1990)E.M.Π. (1990)CR( a ). max
CRCRmax
V. t a
a a
−⎛ ⎞ ⎡ ⎤δ ≈ ⋅ ⋅ ⋅ − ⋅⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎣ ⎦⎝ ⎠
211 15 1
0 080 1
aCR/amax
PE
RM
AN
EN
T D
ISP
LA
CE
ME
NT
(in
)
Newmark - I (1965)
Newmark – II (1965)
Richards & Elms (1979)
Ε.Μ.Π. (1990)
άνω όριογια διάφορα Μ
άνω όριογια διάφορα Μ
Comparison with numerical predictions for actual earthquakes by Franklin & Chang (1977) . . . . Comparison with numerical predictions for actual earthquakes by Franklin & Chang (1977) . . . .
Seismic failure & downslope sliding
Relative Velocity
Relative SlidingRelative Sliding
max max
CRmaxd
max max
CRmax
V a.
aamin
V a.
aa
⎧ ⎫⎛ ⎞⎪ ⎪⋅ ⋅ ⎜ ⎟⎪ ⎪⎝ ⎠δ = ⎨ ⎬⎪ ⎪⎛ ⎞
⋅ ⋅ ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
42
22
0 087
0 50
for EXAMPLE . . . . . .for EXAMPLE . . . . . .
PEAK SEISMIC ACCELERATION amax = 0.50g
PEAK SEISMIC VELOCITY Vmax = 1.00 m/s (Te ≈ 0.80 sec)
“CRITICAL” or “YIELD” ACCELERATION aCR = 0.33g (=2/3 amax)
Relative SlidingRelative Sliding
max max
CRmaxd
max max
CRmax
V a.
aamin
V a.
aa
⎧ ⎫⎛ ⎞⎪ ⎪⋅ ⋅ ⎜ ⎟⎪ ⎪⎝ ⎠δ = ⎨ ⎬⎪ ⎪⎛ ⎞
⋅ ⋅ ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
42
22
0 087
0 50
9 cm !!THUSTHUS,if we can tolerate some small down-slope displacements,the pseudo static analysis is NOT performed for the peak seismicacceleration amax, but for the . . . .
EFFECTIVE seismic acceleration EFFECTIVE seismic acceleration aaEE = (0.50 = (0.50 ÷÷ 0.80) 0.80) aamaxmax
The pseudo static SEISMIC COEFFICIENT khEThe pseudo static SEISMIC COEFFICIENT khE
why is it much lower than the peak seismic acceleration amax ?
why is it much lower than the peak seismic acceleration amax ?
Using the PEAK seismic acceleration (i.e. ) is TOO conservative.
Instead we use the EFFECTIVE seismic acceleration, i.e.
with FSd = 1.0÷1.10
as this will usually lead to fairly small (< 10 cm) downslopedisplacements
= maxh
ak g
= ÷ maxh,E
ak ( . . ) g0 50 0 80
FIRST. . . .FIRST. . . .
SECONDLY . . . . SECONDLY . . . .
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.22g
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
37.82 cm/sec
averagetimehistories
0.22g
average amax=0.60g
observe these numerical results ……
This is because tall earth dams (i.e. H > 30m) are flexible and consequently the seismic motion is NOT synchronous all over the the sliding mass:
For common earth dams Η=(30 ÷ 120m) & earthquakes (Te=0.30 ÷ 0.60s)
λ ≈ (1.00 ÷ 2.00) H i.e.
λ Η
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.22g
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
37.82 cm/sec
Σχήμα 4.2 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
averagetimehistories
0.22g
average amax=0.60g
AS A RESULT OF THESE EFFECTS . . . . AS A RESULT OF THESE EFFECTS . . . .
Kh = 0.22 (from the average acceleration time history)khE=(0.50÷0.80) kh = 0.11 ÷ 0.18
8. 2 Review & Evaluation of
SEISMIC COEFFICIENTS khE
proposed in the literature
REVIEW of khE values, proposed …..
REVIEW of khE values, proposed …..
on the basis of mere engineering . . . INTUITION
in relation with the FREE FIELD peak ground acceleration (PGA)
in relation with the peak seismic acceleration at the DAM CREST
EVALUATION in comparison with numerical analyses which take into account: EVALUATION in comparison with numerical analyses which take into account:
foundation SOIL CONDITIONS
dynamic DAM RESPONSE
NON-LINEAR HYSTERETIC soil response to seismic excitation
Numerical Evaluation of khE :The case of Ilarion Dam in Northern Greece
Vs (m/sec)
1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 9.00E+02 1.00E+03
1700 m/sec
2400 m/sec)
2300 m/sec
1500m/sec 2200 m/sec
1300 m/sec
120 m
Dam geometry &shear wave velocitydistribution.
τ
γ
Basic aspects of numerical modeling
1.21g
0.25g
0.41g
Typical acceleration time histories !!
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.56g
average (amax, i)=0.91g
0 1 2 3 4 5 6 7 8 t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
72,84 cm/sec
Kh = 0.56 (from the average acceleration time history)khE=(0.50÷0.80) kh = 0.28 ÷ 0.45
averagetimehistories
average amax=0.91g
0.56g
Shallow failure surface . . . Shallow failure surface . . .
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.22g
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
37.82 cm/sec
Kh = 0.22 (from the average acceleration time history)khE=(0.50÷0.80) kh = 0.11 ÷ 0.18
averagetimehistories
0.22g
average amax=0.60g
Deep failure surface . . . Deep failure surface . . .
TERZAGHI (1950)0.10 “significant” earthquakes
khE = 0.20 “violent” earthquakes0.50 “destructive” earthquakes
Ad-hoc values of khE based on ENGINEERING EXPERIENCE
0 0.1 0.2 0.3 0.4 0.5PGA (g)
0
0.2
0.4
0.6
0.8
k hE
"significant"
"violent"
"destructive"
0 20 40 60 80 100 120 140H (m)
0
0.2
0.4
0.6
0.8
k hE
"significant"
"violent"
"destructive"
0 0.1 0.2 0.3 0.4 0.5PGA (g)
0
0.2
0.4
0.6
0.8
k hE
STANDARD PRACTICE of the 70’s
khE = 0.10 ÷ 0.20 (depending on M)
FSd > 1.00 ÷ 1.15
0 20 40 60 80 100 120 140H (m)
0
0.2
0.4
0.6
0.8
k hE range of
common design values
BRITISH STANDARDS (Charles et al 1991)
khE ≈ EGA/g EGA
kh,EgH
z
Correlation of khE with the EGA(effective FREE FIELD acceleration)
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
k hE / (E
GA
/ g
)
British standards (Charles et al 1991)
stiff soil or rock
soft soil
EUROCODE EC-8
khE = 0.50 S ST (EGA/g)
S = Soil Factor
1.401.60< 250< 50SHALLOW C or D
E1.351.80< 70< 15< 180D1.201.5070 - 25015 - 50180-360C1.201.35> 250> 50360-800B1.001.00--> 800A
M>5.5M<5.5SCU
(kPa)NSPTVS
(m/s)GroundType
EGA
kh,EgH
z
EGA
kh,E
H
z
EUROCODE EC-8
khE = 0.50 S ST (EGA/g)
ST = Topography Factor (only for H>30m and i>15o)
1.20 ÷ 1.40
1.00
ST
EUROCODE EC-8
khE = 0.50 S ST (EGA/g)
EGA
kh,EgH
z
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
khE /
(EG
A /
g)
range of EC-8
stiff soil or rock
soft soil
GREEK NATIONAL SEISMIC CODE EAK 2000
amax,crest
amax,basePGA
Vs
Te
To=(2.5÷2.8) H/Vs
amax, base= 0.50 PGA
amax, crest= β(Το) amax, base
amax,base
amax,crest
0 0.5 1 1.5 2T (sec)
0
0.5
1
1.5
2
2.5
3
Sa /
am
axgr
A
B
Γ
To
β(Το
)
Η
GREEK NATIONAL SEISMIC CODE EAK 2000
EGA
amax,av
H
z
amax(z/H)
amax,base=0.50 PGA
amax,crest=0.50 β(Το) PGA
max,crest maxmax,av
a a (z / H)a
+=
2hE max,av
hE max,av
k . [( . . ) a / g]
or
k ( . . ) a / g
= ⋅ ÷ ⋅
= ÷ ⋅
0 50 0 50 0 80
0 25 0 40
GREEK NATIONAL SEISMIC CODE EAK 2000
EGA
kh,EgH
z
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
k hE /
(EG
A /
g)
EAK 2000range
stiff soil or rock
soft soil
Correlation of khE with the acceleration at the CREST
amax,crest
kh,Eg
Hz
MARCUSON (1981)
khE = 0.33÷0.50 (amax,crest/g)and
kh = 0.50÷0.75 (amax,crest/g)(kh ≈ 1.50 khE)
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
kh /
(am
ax,c
rest/g
)Marcusson (1981)
range
soft soil
stiff soil or rockhow do yo
u com
pute
a max,c
rest?
MAKDISI & SEED (1978)
khE ≈ 2/3 kh
kh = μ (amax,crest/g)
amax,crest
khEg
Hz
μ
z/H
MAKDISI & SEED (1978)
khE ≈ 2/3 kh
kh = μ (amax,crest/g)
amax,crest
khEg
Hz
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
k h /
(am
ax,c
rest/g
)
Makdisi & Seed (1978)range
soft soil
stiff soil or rock
amax,crest
khEg
Hz
0 20 40 60 80 100 120 140
H (m)
0
0.5
1
1.5
2
2.5
3
k hE(M
& S
) / k
hE(a
nal
yses
)
0.50 + 0.15
stiff soil or rock
soft soil
e.g. from EAK 2000
how do you compute amax,crest ?
Parametric Analysis of NUMERICAL RESULTS
step-by-step METHODOLOGY OUTLINE
EVALUATION OF PROPOSED METHODOLOGYerror margins
EXAMPLE APPLICATIONthe case of Ilarion dam in Northern Greece
8.3 A New Integrated Approach (Papadimitriou & Bouckovalas, 2007)
Towards to a new integrated approach . . . .Towards to a new integrated approach . . . .
PGAb PGA
amax,crest
amax,base
Kh g
To refine kh & khE predictions, we used the results from the numerical analyses to identify the factors affecting:
kh / PGA
kh / amax,crestamax,crest / amax,base
amax,base / PGAamax,crest / PGA
Factors affecting kh
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
kh
g /
PG
A
stiff soil or rock
soft soil
PGAb PGA
amax,crest
amax,base
Kh g
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
kh
g /
PG
A
stiff soil or rock
soft soil
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
kh g
/ a
max
,cre
st
Makdisi & Seed (1978)range
soft soil
stiff soil or rock
0 20 40 60 80 100 120 140
H (m)
0
1
2
3
4
5
6
7
a max
,cre
st /
am
ax,b
ase
stiff soil or rock
soft soil
Factors affecting amax,crest / amax,base
PGAb PGA
amax,crest
amax,base
Kh g
0 0.2 0.4 0.6 0.8 1 1.2
To (sec)
0
1
2
3
4
5
6
7
a max
,cre
st /
a max
,bas
e
stiff soil or rock
soft soil
0 20 40 60 80 100 120 140
H (m)
0
1
2
3
4
5
6
7
a max
,cre
st /
am
ax,b
ase
stiff soil or rock
soft soil
0 1 2 3 4 5 6
To / Te
0
1
2
3
4
5
6
7
a max
,cre
st /
a max
,bas
e
stiff soil or rock
soft soil
0 20 40 60 80 100 120
H (m)
0
0.5
1
1.5
2
a max
,bas
e / P
GA
0.75
soft soil
stiff soil or rock
0 20 40 60 80 100 120
H (m)
0
0.5
1
1.5
2
a max
,bas
e /
PG
A
0.75
soft soil
stiff soil or rock
Factors affecting amax,base / PGA
PGAb PGA
amax,crest
amax,base
Kh g
0 20 40 60 80 100 120
H (m)
0
0.5
1
1.5
2
a max
,bas
e / P
GA
0.75
soft soil
stiff soil or rock
0 20 40 60 80 100 120
H (m)
0
0.5
1
1.5
2
a max
,bas
e /
PG
A
0.75
soft soil
stiff soil or rock
0 0.4 0.8 1.2
To (sec)
0
0.5
1
1.5
2
a max,
bas
e /
PG
A
0.75
stiff soil or rock
soft soil
amax,base / PGA= 0.75+0.25
stiff soil or rock
soft soil
0 0.4 0.8 1.2
To (sec)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
0 20 40 60 80 100 120 140
H (m)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
stiff soil or rock
soft soil
Factors affecting amax,crest / PGA
PGAb PGA
amax,crest
amax,base
Kh g
stiff soil or rock
soft soil
0 0.4 0.8 1.2
To (sec)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
0 20 40 60 80 100 120 140
H (m)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
stiff soil or rock
soft soil
stiff soil or rock
soft soil
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
a max
,cre
st /
PG
A
step-by-stepMETHODOLOGY OUTLINE
step-by-stepMETHODOLOGY OUTLINE
PGAb PGA
amax,crest
kh
STEP 1: Define PGAb and predominant shaking period Te
STEP 2: Compute PGA
STEP 3: Compute predominant dam period To and amax,crest
STEP 4: Compute kh and kh,E
1 23
4
PGAb, Te PGA
HS,VSS
NUMERICALLY (SHAKE, etc.)
OR
APPROXIMATELLY . . . .
20.17b s
eb
22 2
s s
e e
PGA T1 0.85
g TPGA PGA
T T1 1.78
T T
− ⎛ ⎞⎛ ⎞+ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠=⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
1.041.3s b
s ssss
4H PGAT 1 5330 V
V g−⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
STEP 2: Compute PGA . . . .
L
L L
L
H H
HH
rectangular half-circle
triangulartrapezoidal
(A) FUNDAMENTAL DAM PERIOD OF VIBRATION (To) & (To)3-D
STEP 3: Compute TO & amax,crest
oS
H bT ( . r) , r .
V B= + = ≤2 6 2 0 05
VsΗ
b
B
(B) PEAK SEISMIC ACCELERATION AT CREST amax,crest
o o
e e
max,crest o
e
2/3
e o
o e
T T1 4.4 , 0 0.5
T T
a T3.2 , 0.5 2.0
PGA T
2T T3.2 , 2.0
T T
⎧ ⎛ ⎞⎪ + ≤ ≤⎜ ⎟⎪ ⎝ ⎠⎪⎪= ≤ ≤⎨⎪⎪
⎛ ⎞⎪ ≤⎜ ⎟⎪ ⎝ ⎠⎩
STIFF foundation soil or rock
o o
e e
max,crest o
e
2/3
e o
o e
T T1 0.8 , 0 0.5
T T
a T1. , 0.5 2.0
PGA T
2T T1. , 2.0
T T
⎧ ⎛ ⎞⎪ + ≤ ≤⎜ ⎟⎪ ⎝ ⎠⎪⎪= ≤ ≤⎨⎪⎪
⎛ ⎞⎪ ≤⎜ ⎟⎪ ⎝ ⎠⎩
4
4
SOFT foundation soil
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
2.00.5
3.2
1.4
?
?
stiff soil or rock
soft soil
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
k h /
(am
ax,c
rest/g
)
Makdisi & Seed (1978)range
Average value Upper limit
soft soil
stiff soil or rock
khgH
z
amax,crest
Kh,E = (0.50÷0.80) kh
STEP 4: Compute Seismic Coefficients kh & kh,E
h
max,crest
z z1 1.725 , 0.4
H Hk g
a z0.31 , 0.4
H
⎧ ⎛ ⎞ ⎛ ⎞− ≤⎜ ⎟ ⎜ ⎟⎪⋅ ⎪ ⎝ ⎠ ⎝ ⎠= ⎨⎛ ⎞⎪ >⎜ ⎟⎪ ⎝ ⎠⎩
h
max,crest
z z1 1.425 , 0.4
H Hk g
a z0.43 , 0.4
H
⎧ ⎛ ⎞ ⎛ ⎞− ≤⎜ ⎟ ⎜ ⎟⎪⋅ ⎪ ⎝ ⎠ ⎝ ⎠= ⎨⎛ ⎞⎪ >⎜ ⎟⎪ ⎝ ⎠⎩
UPPER BOUND estimates
AVERAGE estimates
EVALUATION OF THE NEW METHODOLOGYerror margins
EVALUATION OF THE NEW METHODOLOGYerror margins
0 20 40 60 80 100 120 140
H (m)
0
0.5
1
1.5
2
2.5
3
kh(p
rop
os
ed
) / k
h(a
na
lys
es
)
1.0 + 0.24
all foundation soils
0 0.2 0.4 0.6 0.8 1
z / H
0
0.5
1
1.5
2
2.5
3
kh(p
rop
ose
d)
/ k h
(an
alys
es)
1.0 + 0.24 all foundation soils
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
kh(p
rop
os
ed
) / k
h(a
na
lys
es
)
1.0 + 0.24
all foundation soils
AVERAGE predictionsAVERAGE predictions
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
kh(p
rop
os
ed
) / k
h(a
na
lys
es
)
1.0 + 0.24
all foundation soils
UPPER BOUND predictionsUPPER BOUND predictions
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
kh(u
pp
er
limit
) / k
h(a
na
lys
es
)
1.32 + 0.33
all foundation soils
AVERAGE predictionsAVERAGE predictions
EXAMPLE APPLICATIONthe case of Ilarion dam in Northern Greece
EXAMPLE APPLICATIONthe case of Ilarion dam in Northern Greece
STEP 1: Define PGAb and predominant shaking period Te
0 4 8 12 16t (sec)
-0.4
-0.2
0
0.2
0.4
x,ac
c (g
)
0 4 8 12 16t (sec)
-60
-40
-20
0
20
40
60
velo
city
(cm
/sec
)
0.37 g
-40.5 cm/sec
ΑΙΓΙΟ - 15/6/1995 - Μs = 6.2(Φράγμα Ιλαρίωνα)
PGAb=0.37g
Te ≈ 0.45s
PGAb=0.37g
Te ≈ 0.45s
0.1 0.3
1
1 2
0
2
3
PERIOD (sec)
Sa /
amax
STEP 2: Compute free field seismic ground acceleration PGA
Vs (m/sec)
1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 9.00E+02 1.00E+03
1700 m/sec
2400 m/sec)
2300 m/sec
1500m/sec 2200 m/sec
1300 m/sec
120 m
PGA=PGAb=0.37gPGA=PGAb=0.37g
The dam is constructed on weathered rockformations and consequently:
STEP 3(A): Compute fundamental dam period TO
Vs (m/sec)
1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 9.00E+02 1.00E+03
1700 m/sec
2400 m/sec)
2300 m/sec
1500m/sec 2200 m/sec
1300 m/sec
120 m
VSD=300-400 m/s
HD=120 m
r=crest/base=0
TO=(2.6+2r) HD/VSD=0.78-1.04 s
TO,3-D = 0.90 TO = 0.70–0.94 s
STEP 3(B): Compute peak seismic acceleration at crest amax,crest
STIFF foundation soil
To/Te = 1.50 ÷ 2.00
0 1 2 3 4 5 6
To / Te
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
am
ax,c
rest /
PG
A
2.00.5
3.2
1.4
?
?
stiff soil or rock
soft soil
amax,crest=3.2 PGA
or
amax,crest=3.2x0.37g=1.18g
1.21g
0.25g
COMPARE WITHNUMERICAL ANALYSIS
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
2 3 4 5 6 7 8t(sec)
average (amax, i)=0.91g
0 1 2 3 4 5 6 7 8 t (sec)
-100
-80
-60
-40
aver
ag
72,84 cm/sec
4.1 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AUH1-2 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
k h /
(am
ax,c
rest/g
)
Makdisi & Seed (1978)range
Average value Upper limit
soft soil
stiff soil or rock
STEP 4: Compute seismic coefficients kh, khE (Shallow failure)
Kh = 0.64 amax,crestorKh = 0.64 x 1.18g=0.75g
khE=(0.50-0.80) kh = 0.38 – 0.60g
for z/H = 0.20
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.56g
average (amax, i)=0.91g
0 1 2 3 4 5 6 7 8 t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
72,84 cm/sec
Σχήμα 4.1 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AUH1-2 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
COMPARE WITHNUMERICAL:RESULTSaveragetimehistories
0.56g
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60av
Σχήμα 4.2 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
0 0.2 0.4 0.6 0.8 1
z / H
0
0.2
0.4
0.6
0.8
1
1.2
k h /
(am
ax,c
rest/g
)
Makdisi & Seed (1978)range
Average value Upper limit
soft soil
stiff soil or rockKh = 0.31 amax,crestorKh = 0.31 x 1.18g=0.37g
khE=(0.50-0.80) kh = 0.18 – 0.30g
for z/H = 0.90
STEP 4: Compute seismic coefficients kh, khE (Deep failure)
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.22g
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
37.82 cm/sec
Σχήμα 4.2 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
0.22gCOMPARE WITHNUMERICALPREDICTIONS
averagetimehistories
for z/H = 0.90
Kh = 0.31 amax,crestorKh = 0.31 x 1.18g=0.37g
The PSEUDO STATIC approach with FSd >1.0 does notprevent slope stability failure.
However,
for STABLE SOILS, it ensures that only very small (e.g.<10 cm) downslope displacements will occur.
for UNSTABLE SOILS (e.g. liquefiable): NEVER USE IT !
8.4 CONCLUDING REMARKS
If we can tolerate these displacements, the SEISMIC COEFFICIENT khE may become much smaller than the corresponding peak seismic acceleration:
PGA
amax,crest
khEgH
z
khE = (0.25 ÷ 1.60) PGA/g
khE = (0.15÷0.56) amax,crest/g
In general, the higher values are associated with • Tall Embankments (H > 30m) &• Shallow failure surfaces (z/H < 0.40)
In general, the higher values are associated with • Tall Embankments (H > 30m) &• Shallow failure surfaces (z/H < 0.40)
**
When ONLY the PGA is known,
you may use:
- the BRITISH STANDARDS (conservative for z/H > 0.40)
- EAK 2000 (O.K. for z/H > 0.40)
The EC-8 is rather UN-conservativeThe EC-8 is rather UN-conservative
amax,crest
khEgH
z
PGA
When the maximum CREST ACCELERATION amax, crest is known (e.g. from seismic analysis of the dam),
you may use:
- MARCUSON (1981) [only for very shallow z/H < 0.30]
- MAKDISHI & SEED (1978) (overconservative for z/H = 0.15 ÷ 0.60)
amax,crest
khEgH
z
PGA
The new INTEGRATED APPROACH provides a rational approach to the computation of the (peak or the effective) seismic coefficient, in terms of the free field seismic motion parameters, as well as the characteristic of the foundation soil and the embankment.
However, note that the method is still under development (more analyses are performed for soft foundation soil and other than trapezoidal embankment sections) and consequently it should be used in parallel with some other recognized approach.
PGAb PGA
amax,crest
amax,base
Kh g
1st HOMEWORK: Earthquake – induced Permanent displacements of an infinite slope
This HWK concerns an idealized geotechnical natural slope, where5m of weathered (soil-like) rock rests on the top of intact rock. The inclination of the slope relative to the horizontal plane is i=25deg, while the mechanical properties of the weathered rock are γ=18κΝ/m3, c=12.5 kPa and φ=28deg. No ground water is present. Assuming infinite slope conditions:
(a) Compute the static factor of safety FSST.
(b) Compute the seismic factor of safety FSEQ., for a maximum horizontal acceleration aH,max=0.45g accompanied by a maximum vertical acceleration aV,max=0.15g.
(c) In case that FSEQ., comes out less than 1.00, compute the associated downslope displacements, for an estimated predominant excitation period Te=0.50 sec.
2nd HOMEWORK:The case of Ilarion Dam in Northern Greece
Vs (m/sec)
1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 9.00E+02 1.00E+03
1700 m/sec
2400 m/sec)
2300 m/sec
1500m/sec 2200 m/sec
1300 m/sec
120 m
The dam is constructed on weathered rockformations:
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
aver
age
acce
lera
tion
(g)
0.22g
average (amax,i) =0.60g
0 1 2 3 4 5 6 7 8t (sec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
aver
age
velo
city
(cm
/sec
)
37.82 cm/sec
Σχήμα 4.2 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AU-8 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)z/H = 0.90
0.39 0.520.87
0.310.620.81
0.230.290.41
0.260.290.88
0.310.470.551.17
0.720.890.691.24
0.58
0 1 2 3 4 5 6 7 8t(sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
aver
age
acce
lera
t
average (amax, i)=0.91g
0 1 2 3 4 5 6 7 8 t (sec)
-100
-80
-60
-40
-20
0
20
aver
age
velo
city
(
72,84 cm/sec
Σχήμα 4.1 : Υπολογισμός μέσης οριζόντιας επιτάχυνσης και ταχύτητας για την επιφάνεια ολίσθησης AUH1-2 (Σεισμική Διέγερση ΑΙΓΙΟ 1995)
Compute seismic coefficients kh, khE
• for the following potential failure surfaces
z/H = 0.20
and the following peak seismic accelerations and predominant
periods for the (horizontal) seismic excitation:
- Low frequency excitation amax=0.37g, Te=0.20 s
- High frequency excitation amax=0.20g, Te=0.65 s