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