9. Seismic Design of RETAINING STRUCTURES - NTUAusers.ntua.gr/.../geot_earthquake_eng_Ch9A-WALLS-SH-ADERS-10.pdf · 9. Seismic Design of RETAINING STRUCTURES GRAVITY WALLS October

9. Seismic Design of

RETAINING STRUCTURES

GRAVITY WALLS

9. Seismic Design of

RETAINING STRUCTURES

GRAVITY WALLS

October 2009

Part A:

G. BOUCKOVALASProfessor of NTUA

CONTENTSCONTENTS

9.1 DYNAMIC EARTH PRESSURES for DRY SOIL

9.2 HYDRO-DYNAMIC PRESSURES

9.3. DYNAMIC PRESSURES for SATURATED SOILS

9.4 PSEUDO STATIC DESIGN

9.5 DISPLACEMENT based DESIGN(performance based design)

Sggested ReadingSggested Reading

Steven Kramer: Chapter 11GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9A.1

9.1 DYNAMIC EARTH PRESSURES for DRY SOIL

The method of ΜΟΝΟΝΟΒΕ - ΟΚΑΒΕ

GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9A.2


9.2 HYDRO-DYNAMIC PRESSURES

WESTERGAARD (1933)

Hydro-STATIC pressures

Hydro-DYNAMIC pressures

applicationpoint: Η/3 from base

applicationpoint: 0.40Η from base

ws w

H

ws ws w

p (x) x

P p (x)dx H

= γ

= = γ∫ 2

0

12

( )

wd h w

wd h w h ws

p (x) k H x / H

P k H . k P

± = γ

± = γ =2

78

71 17

12

REMARKS:

Westergaard theory applies under the following assumptions:

free water (no backfill)

vertical wall face

very large (theoretically infinite) extent of water basin

ATTENTION !The excess pore pressures are positive in front of the wall and negative behind it. Thus the total hydro-dynamic pressure acting on a submerged wall is twice that given by the Westergard solution!


Effect of tank widthEffect of tank widthEffect of tank width

application point: 0.40Η from the base

( )

n

n

n

wd h w

wd h w

h ws

p (x) k H x / H

P k H

.

C

C P

C

k

± = γ

± = γ

=

2

78

712

1 17

n

n

ό

L / HC .

L / H

(C . L / H . )

που

= <+

= για >

41 0

3 1

1 00 2 70

Effect of wall inclinationEffect of wall inclinationEffect of wall inclination

wd h wm

mwd h w

Westergaard

x x x xp (x, ) k ( ) ( )

H H HC ( )

C ( )

H

ή,

xp (x, )

H

ά

k

προσε

⎡ ⎤± α = γ Η − + −⎢α

⎡ ⎤α ⎢ ⎥

γγι

⎥⎣ ⎦

± α = γ Η

ι

⎣ ⎦

στ κ

2 2

78

Zangar (1953) & Chwang (1978)

or, approximately


wd h w

wd h w

h ws

m

m

m

m

rad

xp (x, ) k

H

P k

( . k P )

( )C . ( ) .

C

C

C

ό

ο

και

που

± α = γ Η

± = γ Η

=

α≈ α ≈

π

2

78

7121 17

0 012 2 0

application point: 0.40Η από την βάση

Effect of wall inclinationEffect of wall inclinationEffect of wall inclination

and

where


( )

wd h w

wd h w

h w

e

e s

ep (x,e,..) k H x / H

P (e,..) k H

. k

C

P

C

C

± = γ

± = γ

≈

2

78

712

1 17

we

w

w

6w

ό

n HC . . tanh log

E kT

n ώ

ό βάρος νερού

Η = βάθος νερού

Ε =Μέτρο συμπ. νερού ( 2 10 kPa)

k = συντελεστής διαπερατότητας

Τ = δεσπόζουσα περίοδος δόνησης

που

⎡ ⎤π γ≈ − ⎢ ⎥

⎣ ⎦με= πορ δες

γ = ειδικ

≈ ⋅

220 5 0 5

7

9.3 DYNAMIC PRESSURES for SATURATED FILL

predominant period of shaking

where

withporosity

unit weight of waterwater depth

=Bulk modulus of water (≈2•106kPa)permeability coefficient

Physical analog (Matsuzawa et al. 1985)

WATER + FILLWATERWATER ++ FILLFILL

in other words….

Correction factor Ceexpresses the portion of pore water which vibrates

FREELY,

i.e. independently from the soil skeleton.

Hence, dynamic earth pressures are exerted by the soil skeleton AND the “trapped water”and consequently (you may prove it easily) the Mononobe-Okabe relationships apply for :

γ* = γΞΗΡΟCe+γΚΟΡ.(1-Ce)

Hence, dynamic earth pressures are exerted by the soil skeleton AND the “trapped water”and consequently (you may prove it easily) the Mononobe-Okabe relationships apply for :

γγ* = * = γγΞΗΡΟΞΗΡΟCCee++γγΚΟΡΚΟΡ.(1.(1--CCee))

soil skeleton

“trapped” water, which vibrates together with the soil skeleton

“free” water


EXAMPLE:n=40%, γw=10 kN/m3

Ew=2 106 kPa, T=0.30 sec

Ce > 0.80 pwd ≈ Westergaard

Ce = 0.20÷0.90pwd ≈ Ce·Westergaard

Ce < 0.20 pwd ≈ 0

eH

C . . tanh logk

⎡ ⎤⎛ ⎞= − ⋅ −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

2

0 5 0 5 6 10 6

Fill Material

well graded gravel

gravel

coarse sand

fine sand

silt

Clayey sand & gravel

“Permeable” fill: Cobbles, gravel, Coarse sand (Η<20m)

“Semi-permeable» fill:coarse sand (Η > 20m),fine sand (H < 20m)

“Impermeable” fill:silt, clay, clayey or silty sand and gravel

EXAMPLE:n=40%, γw=10 kN/m3

Ew=2 106 kPa, T=0.30 sec eH

C . . tanh logk

⎡ ⎤⎛ ⎞= − ⋅ −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

2

0 5 0 5 6 10 6


Hydrodynamic pressures on the sea-side of the wall

( )

wd h w

wd h w

h w

m

s

n

m n

m n

p (x) k x / H

P k H

. k

C C

C P

C C

C

= γ Η

= γ

=

2

78

712

1 17

Cm = effect of inclined wallCn = effect of water basin length

Hydrodynamic pressures on the fill-side of the wall

( )

em nwd h w

wd hem n

m e

w

wsn h

p (x) k H x / H

P

C C

C C

C C

k H

. k P

C

C

C

= γ

= γ

=

2

78

712

1 17

Ce = effect of filll

SUMMARY of Hydrodynamic PressuresSUMMARY of Hydrodynamic Pressures

applicati

on

point:

0.40Η from

base


9.49.4 PSEUDO PSEUDO –– STATIC DESIGNSTATIC DESIGN

Genera CaseGenera Case . . . . . .

Wa a

W W

W Wd

h

e e

υδροδυναμικές ωθήσεις (κ ά ύ )

δυναμικές ωθήσεις γαιών

ενεργητική ώθηση γαιών

( k ) *

* C ( C )

P = k ( )H

P H

P P

P

ΚΟΡ

ΞΗΡΟ ΚΟΡ

=

ατ τα προηγο μενα

=ΑΕ γ Η

με γ = γ + − γ

⎡ ⎤= γ − γ⎢ ⎥⎣ ⎦

= γ

Δ = =

⎡ ⎤Δ ⎢ ⎥⎣ ⎦2

2

2

1 3

2 4

1

12

12

active earth pressure

hydrodynamic pressures

=dynamic earth pressures

with γ* = Ce γDRY + (1-Ce)γSAT

( γSA T- γW)H2

ATTENTION !Pa computation requires (γκορ-γw) while ∆ΡΑΕ computation requires γ*. Thus, when it is necessary to compute both Pa and ∆ΡΑΕ with a common unit weight (e.g. ΕΑΚ 2002) you must use:

the buoyant unit weght (γκορ-γw)

a modified seismic coefficient

h hw

*k * k

κορ

γ=

γ − γ

Wa a

W W

W Wd

h

e e




( k ) *

* C ( C )

P = k ( )H

P H

P P

P

ΚΟΡ

ΞΗΡΟ ΚΟΡ

=


=ΑΕ γ Η

με γ = γ + − γ

⎡ ⎤= γ − γ⎢ ⎥⎣ ⎦

= γ

Δ = =

⎡ ⎤Δ ⎢ ⎥⎣ ⎦2

2

2

1 3

2 4

1

12

12




with γ* = Ce γDRY + (1-Ce)γSAT

( γSA T- γW)H2


Clayey sandClayey siltSilty sandClayey or silty gravel

Speci

al cas

e

Speci

al cas

e::

««IMPE

RMEA

BLE

IMPE

RMEA

BLE»»

fillfill Wa a

W W

W Wd

h

e e




( k ) *

* C ( C )

P = k ( )H

P H

P P

P

ΚΟΡ

ΞΗΡΟ ΚΟΡ

=


=ΑΕ γ Η

με γ = γ + − γ

⎡ ⎤= γ − γ⎢ ⎥⎣ ⎦

= γ

Δ = =

⎡ ⎤Δ ⎢ ⎥⎣ ⎦2

2

2

1 3

2 4

1

12

12




with γ* = γSAT (Ce=0)

( γSA T- γW)H2

W

W

W Wd

h

h

υδροδυναμικές ωθήσεις =


( k )

(k )(

P P

P

ή

P )

0

ΚΟΡ

ΚΟΡ

ΚΟΡ

ΚΟΡ

=

=

=ΑΕ

=ΑΕ

γ Η

γ − γ Η

Δ = =

⎡ ⎤Δ ⎢ ⎥⎣ ⎦

γ⎡ ⎤Δ ⎢ ⎥γ − γ⎣ ⎦

2

2

1 3

2 4

3

8

kkhh* * ≈≈ 22.2.2 kkhh

Speci

al cas

e

Speci

al cas

e::

««IMPE

RMEA

BLE

IMPE

RMEA

BLE»»

fillfill

Clayey sandClayey siltSilty sandClayey or silty gravel

= hydrodynamic pressures = 0

= dynamic earth pressures

or


W

.

a a

W W

W Wd

h

e

υδροδυναμικές ωθήσεις



( k ) *

*

P = k ( )H

P H

P P

P

(C )

0

ΚΟΡ

ΞΗΡ

==ΑΕ γ Η

με

⎡ ⎤= γ − γ⎢ ⎥⎣ ⎦

= γ

Δ = =

⎡ ⎤Δ ⎢ ⎥

γ

⎦

=

≠

⎣

γ =

2

2

2

1 3

2 4

12

12

1

sandsand & gravelcobblesballast

Speci

al cas

e

Speci

al cas

e::

««PER

MEABL

E

PERM

EABL

E»»fillfill


= hydrodynamic pressure

= dynamic earth pressure

with γ* = γDRY

W

W

W Wd

h

h

υδροδυναμικές ωθήσεις


( k )

(k )(

P P

P

ή

P )

0

ΚΟΡ

ΞΗΡ

ΞΗΡ

ΚΟΡ

=

=

=ΑΕ

=ΑΕ

γ Η

γ − γ Η

Δ = =

⎡ ⎤Δ ⎢ ⎥⎣ ⎦

⎡ ⎤γΔ ⎢ ⎥γ

≠

− γ⎣ ⎦

2

2

1 3

2 4

3

8

kkhh* * ≈≈ 1.61.6 kkhh

Speci

al cas

e

Speci

al cas

e::

««PER

MEABL

E

PERM

EABL

E»»fillfill

sandsand & gravelcobblesballast

= hydrodynamic pressure

= dynamic earth pressure

or


EXAMPLE: What do I do when I am not sure about the permeability of the fill material?

Vertical & smooth wallBasin of infinite length Cm = Cn = 0

Fill:

γΞΗΡΟ=16 kN/m3

γΚΟΡ. = 20 kN/m3

Ce = 0 ÷ 1.0

W Wd h w

*h w

e eh h

w

C ( C )k *

P P k H

P ( k )( )

kΞΗΡΟ ΚΟΡ

κορ

κορ

=ΑΕ

γ + − γμε =

Δ = = γ

Δ γ − γ Η

γ − γ

2

2

1

712

1 32 4

Total horizontal thrust:

ΣFd = ∆ΡΑΕ+Ρwd+CePwd

Total overturning moment:

ΣΜd=0.60H ∆ΡΑΕ+0.40Η (1+Ce) Ρwd

Total horizontal thrust:

ΣFd = ∆ΡΑΕ+Ρwd+CePwd

Total overturning moment:

ΣΜd=0.60H ∆ΡΑΕ+0.40Η (1+Ce) Ρwd

permeable fill

impermeable fill


9.5 DISPLACEMENT based DESIGN(or “performance based design”)

RICHARDS-ELMS METHOD

δ: friction angle between wall side and fillφο: friction angle between wall base and ground

( )o oAE A

o oA AE w w h

N W P P tan

F N tan

N tanF.S.

P P P P k W

ο

ολ ο

= + Δ + δ

= ϕ

ϕ=

+ Δ + + Δ +

Performance based design:Even though F.S.oλ< 1.0 (sliding failure) there is no collapse of the wall (!!),but development of limited displacements, which may be tolerable ……..

SLIDING FAILURE OF GRAVITY WALLS

αcr=Ng : critical seismic acceleration leading to F.S.oλ=1.00


( )cr. max cr

max cr

crmax

V. f t

gμε:

. έδαφος f

. βράχος

−α− αδ = ⋅ ⋅

α α

Να =

α

⎧= ⎨⎩

121 15 1

0 013

6 302 50

Εναλλακτικά:

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)

max

max

.2

4CR

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

−⎛ ⎞ ⎡ ⎤⎜ ⎟≈ ⋅ ⋅ ⋅ − ⋅⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9A.15

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)

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

Relative SlidingRelative Sliding42

max max

CRmaxd 22

max max

CRmax

V a0.087

aaδ min

V a0.50

aa

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

DISPLACEMENT BASED DESIGN(or performance based design)

New design philosophy:

*max h

max h

/** max maxh h

max

V k.

k

Vk k .

g

−⎛ ⎞

δ = ⎜ ⎟α ⎝ ⎠

⎡ ⎤α= = ⎢ ⎥α δ⎣ ⎦

42

1 42

0 087

0 087

Instead of designing the wall for kh=amax/g,I choose a lower kh* (< kh)

which is a function of the allowable wall displacement δ. In that case, the required factor of safety is F.S.=1.00

alternatively:

* hh

w

w /

max

max

kk

q

με: qV

.

=

=⎡ ⎤⎢ ⎥

α δ⎢ ⎥⎣ ⎦

1 42

1

0 087GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9A.17

In accordance with this design philosophy, ΕΑΚ requests that:

γn=importance coefficient

nh

w

max

kq

g

α ⋅ γ=

αα =

* hh

w

kk

q

⎛ ⎞=⎜ ⎟

⎝ ⎠

qw=

2.00 δ(mm)=300a

1.50 δ(mm)=200a

1.25 δ(mm)=100a (τοίχοι από Ο.Σ.)

1.00 anchored flexible walls

0.75 basement walls, etc


Documents

9. Seismic Design of RETAINING STRUCTURES - NTUAusers.ntua.gr/.../geot_earthquake_eng_Ch9A-WALLS-SH-ADERS-10.pdf · 9. Seismic Design of RETAINING STRUCTURES GRAVITY WALLS October