Seismic Analysis of SlopesSeismic Analysis of Slopes

George D. George D. BouckovalasBouckovalasAchilles PapadimitriouAchilles Papadimitriou

National Technical University of AthensSchool of Civil EngineeringGeotechnical Division

April 2007

Current Design PracticeCurrent Design Practice

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

some times we tend to some times we tend to ……. forget F. forget FVV

8.1 The “Pseudo Static” approach: BASIC CONCEPTS

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

FSd > 1 safe conditions

FSd < 1 slope failure (dynamic) ?

FSd > 1 safe conditions

FSd < 1 slope failure (dynamic) ??

8.2 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)

.

.

.

( )2 CRmax

2max CR

1 aVδ 0.50

a a

−⎛ ⎞⎜ ⎟= ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

2max

2max CR

V 1δ 0.50

a a

⎛ ⎞⎜ ⎟≈ ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

RICHARDS & ELMS (1979)RICHARDS & ELMS (1979)

2max

4max CR

V 1δ 0.087

a a

⎛ ⎞⎜ ⎟≈ ⋅ ⋅⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

E.M.Π. (1990)E.M.Π. (1990)CR

2(1 a )1.15 max

CR

CRmax

V 1δ 0.080 t 1 a

a a

−⎛ ⎞ ⎡ ⎤⎜ ⎟≈ ⋅ ⋅ ⋅ − ⋅⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

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 Sliding42

max max

CRmaxd 22

max max

CRmax

V a0.087

aaδ min

V a0.50

aa

⎧ ⎫⎪ ⎪⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⋅ ⋅⎜ ⎟⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⋅ ⋅⎜ ⎟⎪ ⎪⎜ ⎟⎪ ⎪⎜ ⎟⎝ ⎠⎪ ⎪⎪ ⎪⎩ ⎭

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 Sliding42

max max

CRmaxd 22

max max

CRmax

V a0.087

aaδ min

V a0.50

aa

⎧ ⎫⎪ ⎪⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⋅ ⋅⎜ ⎟⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪⎛ ⎞⎪ ⎪⎜ ⎟⎪ ⎪⋅ ⋅⎜ ⎟⎪ ⎪⎜ ⎟⎪ ⎪⎜ ⎟⎝ ⎠⎪ ⎪⎪ ⎪⎩ ⎭

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

8.3 The pseudo static SEISMIC COEFFICIENT khE

why is it much lower than the peak seismic acceleration amax ?

8.3 The pseudo static SEISMIC COEFFICIENT khE

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 ( 0.50 0.80 ) g= ÷

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

8.4 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

Numerical Evaluation of khE : The case of Ilarion Dam in Northern Greece

Basic aspects of numerical modeling

1.21g

0.25g

0.41g

Numerical Evaluation of khE : The case of Ilarion Dam in Northern Greece

Typical acceleration time histories !!

Numerical Evaluation of khE : the case of Ilarion Dam in Northern Greece

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

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

Numerical Evaluation of khE : The case of Ilarion Dam in Northern Greece

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

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 /

(EG

A /

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

k hE /

(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

2

+=

hE max,av

hE max,av

k 0.50 [( 0.50 0.80 ) a / g ]

or

k ( 0.25 0.40 ) a / g

= ⋅ ÷ ⋅

= ÷ ⋅

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

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

0 20 40 60 80 100 120 140

H (m)

0

0.5

1

1.5

2

2.5

3

k hE(E

AK

) / k

hE

(anal

yse

s)

0.85 + 0.28

stiff soil or rock

soft soil

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(E

AK

) / k

hE (a

nal

yses

)

stiff soil or rock

soft soil

GREEK NATIONAL SEISMIC CODE EAK 2000

EGA

kh,EgH

z

0 0.2 0.4 0.6 0.8 1 1.2

To (sec)

0

0.5

1

1.5

2

2.5

3

k hE(E

AK

) / k

hE (a

nal

yses

)

0.85 + 0.28

soft soil

stiff soil or rock

GREEK NATIONAL SEISMIC CODE EAK 2000

EGA

kh,EgH

z

0 1 2 3 4 5 6

To / Te

0

0.5

1

1.5

2

2.5

3

k hE(E

AK

) / k h

E (a

nal

yses

)

0.85 + 0.28

soft soil

stiff soil or rock

GREEK NATIONAL SEISMIC CODE EAK 2000

EGA

kh,EgH

z

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

how do you compute amax,crest ?

amax,crest

khEg

Hz

MAKDISI & SEED (1978)

khE ≈ 2/3 kh

kh = μ (amax,crest/g)

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

amax,crest

khEg

Hz

MAKDISI & SEED (1978)

khE ≈ 2/3 kh

kh = μ (amax,crest/g)

0 0.2 0.4 0.6 0.8 1

z / H

0

0.5

1

1.5

2

2.5

3

kh

E(M

& S

) / k

hE(a

naly

ses

)

stiff soil or rock

soft soil

0.50 + 0.15

e.g. from EAK 2000

amax,crest

khEg

Hz

MAKDISI & SEED (1978)

khE ≈ 2/3 kh

kh = μ (amax,crest/g)

0 0.2 0.4 0.6 0.8 1 1.2

To (sec)

0

0.5

1

1.5

2

2.5

3

k hE(M

& S

) / k h

E(a

nal

yse

s)

0.50 + 0.15

soft soilstiff soil or rock

e.g. from EAK 2000

amax,crest

khEg

Hz

MAKDISI & SEED (1978)

khE ≈ 2/3 kh

kh = μ (amax,crest/g)

0 1 2 3 4 5 6

To / Te

0

0.5

1

1.5

2

2.5

3

kh

E(M

& S

) / k

hE(a

naly

ses

)

soft soil stiff soil or rock

0.50 + 0.15

e.g. from EAK 2000

8.5 C O N C L U D I N G

R E M A R K S . . . . .

CONCLUDING REMARKS . . . .

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 !

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

HOMEWORK 8.1: 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.

HOMEWORK 8.2: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