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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
Active Earth Pressure
LATERAL EARTH
PRESSURE
aa K 2c- K z
z
z o
a K 2c-
Active pressure distribution
Active Earth Pressure
a K 2c-
K za
g
LATERAL EARTH
PRESSURE
Active pressure distribution
Active 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
Active pressure distribution
Active Earth Pressure
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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