CV2302 - Lateral Earth Pressure

WKS - January 2010

CV2302 - Lateral Earth Pressure 1

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CV2302 – Geotechnical Engineering

Lateral Earth Pressure

At-rest Lateral earth pressure

Rankine’s lateral earth pressure theory

Coulomb’s lateral earth pressure theory

Log spiral method of lateral earth pressure

Reference: Craig (2004), pp 161-181

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Lateral Earth Pressure

Retaining Walls

s'z and sz

s'x and sx

In-Situ

WKS - January 2010


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At-Rest Earth Pressure

Coefficient of lateral earth pressure at-rest, Ko

Ko = s'x / s'z

Ko = (1 – sin f') OCR sin f‘ or

Horizontal Stress

Effective: s'x = Ko s'z

Total: sx = s'x + u

Applicable to all sand and clay.s'z and sz

s'x and sx

Ko is applicable to any soil deposit.

Ko = (1 – sin f') OCR 0.5

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At-Rest Earth Pressure

H

s'x = Ko s'z

H/3Po = ½ Ko g H

2

Ko g H

Ko is also applicable to soil acting on a non-yielding wall.

Soil Type Ko

Loose sand 0.6

Dense sand 0.35

NC clay 0.6

OC clay 1 to 2

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Lateral Earth Pressure Theories

• Rankine

• Coulomb

• Log-Spiral

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Rankine’s Earth Pressure Theory

1. The soil is homogeneous and isotropic.

2. Failure surface is a plane.

3. Ground surface is a plane.

4. Wall is infinitely long i.e. plane strain condition.

5. Sufficient wall movements to develop active or passive state.

6. Wall is vertical.

7. Wall is perfectly smooth.

8. The resultant force is parallel to ground surface.

9. The resultant force is acting at H/3 above the wall bottom.

H/3

H

Assumptions

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Active Earth Pressure in Sand (Rankine)

s'x = Ko s'z

No wall movement

Wall moved away from soil

s'A = KA s'zHere we are dealing with effective stress only.

f'

s'zs'xs'A s

t

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Active Earth Pressure in Sand (Rankine)

Failure plane at active state

s'A = KA s'z

Active earth pressure

45 + f'/2

Sin f' = ------- = ----------------AB ½ (s'z – s'A)

OA ½ (s'z + s'A)

KA = -----------------(1 – sin f')

(1 + sin f')

tf'

s'zs'A s

45 + f'/2

O

B

A

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Passive Earth Pressure in Sand (Rankine)

s'P = KP s'z

s'x = Ko s'z

No wall movement

Wall moved towards soil

Here we are dealing with effective stress only.

f'

s'zs'x s'P s

t

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Passive Earth Pressure in Sand (Rankine)

Failure plane at passive state

s'P = KP s'z

Passive earth pressure

45 - f'/2

Sin f' = -----------------½ (s'P - s'z)

½ (s'P + s'z)

KP = -----------------(1 + sin f')

(1 - sin f')

f'

s'z s'P s

t

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Active Earth Pressure in Clay (Rankine)

(under short term undrained condition)

= Kos'z + u

No wall movement

Wall moved away from soil

sA = sz – 2 cu

sx = s'x + u

(-)

(+)

Here we are dealing with total stress.

fu=0

szsxsA s

t

cu

cu = ½ (sz - sA)

sA = sz - 2cu

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Passive Earth Pressure in Clay (Rankine)

(under short term undrained condition)

Here we are dealing with total stress.

fu=0

szsx sP s

t

cu

sP = sz+ 2cu

cu = ½ (sP - sz)

sP = sz + 2cu

= Kos'z + u

No wall movement

Wall moved towards soil

sx = s'x + u

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KA = -----------------(1 – sin f')

(1 + sin f')

Active & Passive Earth Pressure in Clay (Rankine)

(under long term drained condition c' and f' )

t

f'

s'zs'A s

c'

Active earth pressure

s'A = KAs'z – 2 c'√KA

(-)

(+)

Passive earth pressure

s'P = KPs'z + 2 c'√KP KP = -----------------(1 + sin f')

(1 - sin f')

Here we are dealing with effective stress only.

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Lateral Earth Pressure for c- f soil (Rankine)

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Active Earth Pressure for c- f soil (Rankine)

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Passive Earth Pressure for c'- f' soil (Rankine)

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Rankine’s Earth Pressure for c'- f' soil

s'A = KAs'z – 2 c'√KAs'P = KPs'z + 2 c'√KP

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Lateral Earth Pressure with Inclined Backfill (Rankine)

PA = ½ KA g H2

PAV = ½ KA g H2 sin b

PAH = ½ KA g H2 cos b

Active

PA

PAH

PAV

b

b

sand

cos b

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Rankine’s Earth Pressure at Any Point

Clay (drained)

Clay (undrained)

Sand (drained)

PassiveActive

s'A = KA s'z s'P = KP s'z

sA = sz – 2 cu sP = sz + 2 cu

s'A = KAs'z – 2 c'√KAs'P = KPs'z + 2 c'√KP

cos b

b

s'A

s'AH

b

s'AV

KA = -----------------(1 – sin f')

(1 + sin f')KP = -----------------

(1 + sin f')

(1 - sin f')

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Coulomb’s Earth Pressure Theory

1. The soil is homogeneous and isotropic.

2. Failure surface is a plane.

3. Ground surface is a plane.

4. Wall is infinitely long i.e. plane strain condition.

5. Sufficient wall movements to develop active or passive state.

6. Wall can be inclined (a 0).

7. Wall can be rough (d 0).

8. The resultant force is acting at an angle d with the normal to

wall.

l

b

dPA

H/3

H

Assumptionsa

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Coulomb’s Active Earth

Alternatively, ∂P/∂q = 0 and solve for P.

Each wedge will yield one P value.

Try different wedges to find the highest P.

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Coulomb’s Active Earth Pressure

PA

PAH

PAV

l

b

d

PA

H/3

H

PA= ½ KA g H2

PAH = PA cos (d+ l)

PAV = PA sin (d+ l)

a

l = a – 90o

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Coulomb’s Passive Earth Pressure

PP

PPH

PPV

l

b

d

PPH/3

H

PP= ½ KA g H2

PPH = PP cos (d - l)

PPV = PP sin (d - l)

a

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Log Spiral Method on Lateral Earth Pressure

PA= ½ KA g H2

PAH = PA cos d

PAV = PA sin d

PP= ½ KP g H2

PPH = PP cos d

PPV = PP sin d

Active Case Passive Case

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Log-Spiral Method – KA & KAH

KA = KAH / cos d KAV = KAH tan d

KAH

Design Value of f'

d/f’=0

d/f’=0.66

d/f’=1.0

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Log-Spiral

Method

KPH

KPH

Design Value of f'

PPV

PPH

PP

d

d / f' =1.0

d / f' = 0.66

d / f' = 0

Horizontal surface b=0

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Active and Passive Pressure with Wall Adhesion cw

∂P/∂q = 0

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Active Pressure Passive Pressure

Drained case: c’>0 and f’>0 c = c’

Undrained case: cu>0 and fu=0 c = cu Ka = Kp = 1

sa sp

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Active Pressure Passive Pressure

Undrained case: cu>0 and fu=0

If cw/cu = 0 sa = sz – 2 cu sp = sz + 2 cu

If cw/cu = 0.5 sa = sz – 2.45 cu sp = sz + 2.45 cu

sa sp

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Theory Roughness Inclination Application

Rankine smooth vertical KA

Coulomb any any KA

Log-Spiral any any KA & KP

Summary

WKS - January 2010


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Comparisons of Method of Analysis

Theory f' d KA PA PAH

Loose Sand

Rankine 30 0 0.33 83.3 83.3

Coulomb 30 20 0.30 75 70.5

Log-Spiral 30 20 0.32 80 75.2

Dense Sand

Rankine 40 0 0.22 55 55

Coulomb 40 30 0.20 50 43

Log-Spiral 40 30 0.22 55 48

5 mPAH

g = 20 kN/m3

PA

d

PAV

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Comparisons of Method of Analysis

Theory f' d KP PP PPH

Loose Sand

Rankine 30 0 3 750 750

Coulomb 30 20 6.1 1525 1433

Log-Spiral 30 20 5.2 1300 1222

Dense Sand

Rankine 40 0 4.6 1150 1150

Coulomb 40 30 24.9 6225 5391

Log-Spiral 40 30 13.1 3275 2836

5 mPPH

g = 20 kN/m3

PP

d

PPV

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Total active earth pressure on wall in sand

q

Sand, g

H

s'A= KAs'z

s'A= KA (q + gH)

s'A= KAq

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Total active earth pressure on wall in sand

s'A= KAs'z u

Earth pressure

+ surchargeWater

pressure

+

q

Sand

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Total active earth pressure on wall in clay

q

Earth pressure

Clay cu

z

sA= sz – 2cu

where sz = g z + q

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q

Earth pressure

Clay cu

sA= sz – 2cu

where sz = g z + q

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Earth pressure

Clay cu

How do you determine zc?

sA= g z – 2cu

+

-

Ignore negative pressure

Tension crackzcz

Dry condition

sA= g zc – 2cu = 0


Clay cu

How do you determine zw?

After heavy downpour

Final crack depth

Earth pressure

sA= sz – 2cu

zw

Initial crack

depth

sA= g zw – 2cu = zw gw

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Total active earth pressure on wall in layered soil

s'A= KAs'z

u

Earth pressure

+ Surcharge

Water

pressure

+

q

Sand

Clay

sA= sz – 2cu

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Lateral movement

K

KO

KP

KA

at-rest

passive

active

Effect of soil movement on lateral earth pressure in Sand

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Movements required to mobilise active pressure

Soil Type D H = 3 m H = 5 m

Dense sand 0.001H 3 mm 6 mm

Loose sand 0.004H 12 mm 24 mm

Stiff clay 0.010H 30 mm 50 mm

Soft clay 0.020H 60 mm 100 mm

Movements required to mobilise passive pressure

Soil Type D H = 3 m H = 5 m

Dense sand 0.020H 60 mm 100 mm

Loose sand 0.060H 180 mm 300 mm

Stiff clay 0.020H 60 mm 100 mm

Soft clay 0.040H 120 mm 200 mm

Documents

CV2302 - Lateral Earth Pressure