Foundation Engineering_Earth Pressure

8/17/2019 Foundation Engineering_Earth Pressure

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010Foundation Engineering(3-1-0-4)

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Course instructor

Dr. Trudeep N. DaveInstitute of Infrastructure Technology Research and Management

E-mail: [email protected]

Class timings:

Monday: 11:00 to 12:00Tuesday: 10.00 to 11.00

Thursday: 11.00 to 12.00

Retaining wall for embankment

Box culvert for vehicular traffic

Retaining wall with Box culvert

Basement wall


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Bridge AbutmentPedestrian Underpass

At-rest / Active / Passive Earth Pressures

Granular Soils

smooth wall

Wall moves away

from soil

Wall moves

towards soil A

B

Earth pressure at rest

zσv

σh = K o σv

A

B

If wall AB remains sta tic – soi l

mass will be in a state of elas tic

equil ibrium – horizontal strain is

zero.

Ratio of horizontal stress to vertical

stress is called coefficient of earth

pressure at rest, K o, or

v

ho

K

z K K ovoh

LATERAL EARTH

PRESSURE


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Soil with Cohesion and Friction

Mohr’s Circle of

Stress

c

c

f

Soil fails when

Mohr’s circle

touches theselines

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σX = Ko σz

σz

ACTIVE EARTH PRESSURE (RANKINE’S)

(in simple stress field for c=0 soil)

Active Earth Pressure

- in granular soils

v’

decreasing h’

Initially (K0 state)

Failure (Active

state)

As the wall moves away from the soil,

active earth

pressure= pa

σzKo σzσx’Aø


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PASSIVE EARTH PRESSURE (RANKINE’S)

(in simple stress field for c=0 soil)

σX = Ko σz

σz

Passive Earth Pressure- ingranular soils

v’

Initially (K0 state)

Failure (Active

state)

As the wall moves towards the soil,

increasing h’

passive

earthpressure

= pa

S h e a r s t r e s s

Normal

stress

f tanc f

C

D

D’

O A σpKoσv

b

a

σv

f

c

Mohr’s circlerepresenting

Rankine’spassive state.

Passive Earth Pressure


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Based on the diagram :

pressureearthactives Rankine' of t coefficien Ratiov

a

a K (K a is the ratio of the effective stresses)

Therefore :

f

f

sin1

sin-1 )

2 (45 -tan K 2

v

aa

It can be shown that :

aa

2

a

K 2c- K z

)2

(45 -tan2c- )2

(45 -tan zf


LATERAL EARTH

PRESSURE

aa K 2c- K z

z

z o

a K 2c-

Active pressure distribution


a K 2c-

K za

g

LATERAL EARTH

PRESSURE



Based on the previous slide, using

similar triangles show that :

a

o K

c z g

2 where z o is depth of tension

crack

For pure cohesive soil, i.e. when f = 0 :

g

c z

o

2

LATERAL EARTH

PRESSURE

For cohesionless soil,

c = 0

aava K z K

z



K za

g

LATERAL EARTH

PRESSURE


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Earth pressure at rest may be obtained theoretically from thetheory of elasticity applied to an element of soil,

remembering that the lateral strain of the element is zero.

Various researchers proposed empirical relationships for K 0,

EARTH PRESSURE THEORIES

The magnitude of the lateral earth pressure is evaluated

by the application of one or the other of the so-called ‘lateralearth pressure theories’ or simply ‘earth pressure theories’.

Theories given by Coulomb (1776) and Rankine (1857) stood the

test of time and are usually re fer red to as the “Classical earth

pressure theories”.

These theories have been developed originally to apply to

cohesionless soil backfill, since this situation is considered to be

more frequent in practice and

since the designer will be on the safe side by neglecting cohesion.

RANKINE’S THEORY

Rankine (1857) developed his theory of lateral earth pressure

when the backfill consists of dry, cohesionless soil. The theory

was later extended by Resal (1910) and Bell (1915) to be

applicable to cohesive soils.

The following are the important assumptions in Rankine’s theory:

(i ) The soil mass is semi infinite, homogeneous, dry and

cohesionless.(ii ) The ground surface is a plane which may be horizontal or

inclined.

(iii ) The face of the wall in contact with the backfill is vertical

and smooth. In other words, the friction between the wall and

the backfill is neglected (This amounts to ignoring the presence

of the wall).

(iv ) The wall yields about the base sufficiently for the active

pressure conditions to develop; if it is the passive case that is

under consideration, the wall is taken to be pushed sufficiently

towards the fill for the passive resistance to be fully mobilised .


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Plastic Equilibrium of Soil—Active and Passive RankineStates

A mass of soil is said to be in a state of plastic equilibrium if

failure is incipient or imminent at all points within the mass.

This is commonly referred to as the ‘general state of plastic

equil ibrium’ and occurs only in rare instances such as when

tectonic forces act.

Most of the times only in a small portion of the mass such as

that produced by the yielding of a retaining structure in the soil

mass adjacent to it. Such a situation is referred to

as the ‘local state of plastic equilibrium’.


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Rotation of Frictionless Wall About the Bottom

Rankine vs. Coulomb Theory

Coulomb’s Earth Pressure Theory

Coulomb’s wedge theory of earth pressure is based on the concept of a slidingwedge whichis torn off from the restof the backfill on movementof the wall.

Assumptions:

The backfil l is dry, cohesionless, homogeneous, isotropic and elast icallyundefomable but breakable.

Theslip surface is plane which passess throughthe heel of thewall.

The sliding wedge itself acts as a rigid body and the value of earth pressure isobtained by considering the limiting equil ibrium of the sliding wedge as awhole.

Theposition and direction of theresultant earth pressure areknown.

The back of the wall is rough and a relat ivemovement of the wall and the soildevelops frictionalforcesthat influencethe direction of the resultantpressure

Coulomb: Active Case


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Coulomb: Passive Case

Rebhann’s Graphical Method for Active Pressure

1. Draw the ground line and f – line at angles b and f, respectively to meetin pointD.

2. Draw semi-circle on BD as diameter.

3. Through B,draw a l ine BH at an angle ψ with BD. Line BH is called theearth pressure lineor ψ – line.

4. ThroughA, draw lineAE parallel to theψ – line.5. Draw EF perpendicular to BD, to meet thesemi-circlein F.

6. WithB ascentre,and BF asradius,draw an arc to cut BDin G.

7. Through G, draw GC parallel to the ψ – line. BC then represents the slipline.

8. With G asthe centre and GCas radius, draw an arc to cut BD in L.JoinCL.

9. Calculate the total active earth pressure from therelation:

Pa = g(∆CLG)= ½ g (LG) * x


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Culmann’s Graphical Solution

A graphical solution of Coulomb’s earth-pressure theory by Culmann(1875).

Culmann’s solution can be used for any wall fr iction, regardless o f

irregularity of backfill and surcharges.

Steps in Culmann’s solution of active pressure with

granular backfill

Step 1: Draw the features of the retaining wall and the backfill to a convenient scale.

Step 2: Determine the value of y (degrees) = 90 – q – d ’, where, q the inclination of

the back face of the retaining wall with the vertical,and d ’ = angle of wall friction.

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Step 3: Drawa line BDthatmakesan angle f‘ with the horizontal.

Step 4: Drawa line BE thatmakesan angle y with lineBD.

Step 5: To consider some trial failure wedges, draw lines BC1, BC2, BC3, . . . ,BCn.

Step 6: Find the areas ofABC1,ABC2,ABC3, . . . ,ABCn.

Step 7: Determine the weight o f soil, W, per unit length of the retaining wall ineach of the trial failure wedges as follows:

W1 = (Area of ABC1) × (g) × (1)

W2 = (Area of ABC2) × (g) × (1)

W3 = (Area of ABC3) × (g) × (1)

Wn = (Area of ABCn) × (g) × (1)

Step 8:Adopt a convenient load scale and plot the weights W1, W2, W3, . . ., Wndetermined from step 7 on line BD. (Note: Bc1 = W1, Bc2 = W2, Bc3 = W3, .. . , Bcn = Wn.)

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Step 9: Draw c1c1’, c2c2’, c3c3’, . . . , cncn’ parallel to the line BE. (Note: c1’,c2’, c3’, . . . , cn’ are located on lines BC1, BC2, BC3, . . . , BCn, respectively.)

Step 10: Draw a smooth curve through points c1’, c2’, c3’, . . . , cn’, Thiscurve is called the Culmann line.

Step 11: Draw a tangent B’D’ to the smooth curve drawn in Step10. B’D’ isparallel to line BD. Let c’a be the point of tangency.

Step 12: Draw a line ca c’a parallel to the line BE.

Step 13: Determine the active force per unit length of wall asPa = (Length of cac’a) × (Load scale)

Step 14: Draw a line Bc’aCa .ABCa is the desired failure wedge.


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Foundation Engineering_Earth Pressure

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