6.1 general

6.1 General

Conditions of plane strain are assumed, i.e. strains in the longitudinal direction of the structure are assumed to be zero. It is assumed that the stress–strain behavior of the soil can be represented by the rigid–perfectly plastic idealization.

E

Dock Tunnel

EE

Abutment

E

Lateral Earth Pressure

加筋土挡墙

挖孔桩支护

钢筋混凝土挡土墙


完工

完工


钢筋混凝土挡土墙

Rigid earth retaining wall

扶壁式

刚性加筋

预应力

板桩变形

板桩上土压力实测

计算

锚杆

板桩

Flexible earth retaining wall


retaining and protection of foundation excavation


• There are 3 states of lateral earth pressureKo = At Rest

Ka = Active Earth Pressure (wall moves away from soil)

Kp = Passive Earth Pressure (wall moves into soil)

Passive is more like a resistance


Relation among three earth pressures

-△ +△

+△-△

E

o△a △p

Ea

Eo

Ep


6.2 At Rest Earth Pressure

One common earth pressure coefficient for the “at rest” condition in granular soil is:

Ko = 1 – sin(φ)

Where: Ko is the “at rest” earth pressure coefficient and φ is the soil friction value.

K0h

h

z

K0z

oo KhE 2

2

1

z

h/3


Retaining wall between the slope


Failured earth retaining wall


Failured reinforced earth retaining wall


6.2 Rankine Theory of Earth Pressure

If the soil mass as a whole is stressed such that the principal stresses at every point are in the same directions, theoretically, there will be a network of failure planes equally inclined to the principal planes.

The Rankine Theory assumes: There is no adhesion or friction between the wall and soil Lateral pressure is limited to vertical walls Ground surface is horizontal.


pa pp

f

zK0z

f =c+ tan

伸展压缩45o-/2 45o ＋ /2

If there is a movement of the wall away from the soil, the value of x decreases as the soil dilates or expands outwards, the decrease in x being an unknown function of the lateral strain in the soil. If the expansion is large enough, the value of x decreases to a minimum value such that a state of plastic equilibrium develops. Since this state is developed by a decrease in the horizontal stress x, this must be the minor principal stress ( 3). The vertical stress z is then the major principal stress ( 1).


1) Active Earth Pressure

45o ＋ /2

h

z(σ1)

pa(σ3)

z

245tan2

245tan2

13

oo c

aaa KczKp 2

When the horizontal stress becomes equal to the passive pressure the soil is said to be in the passive Rankine state, there being two sets of failure planes each inclined at 45o ＋ /2 to the vertical (the direction of the major principal plane)


For cohesive soil

This means that in active case the soil is in a state of tension between the surface and depth z0. In practice, however, this tension cannot be relied upon to act on the wall, since cracks are likely to develop within the tension zone and the part of the pressure distribution diagram above depth z0 should be neglected.

For sandy soil

h/3

Ea

hKa

aa zKp

h

aKh2)2/1(


2) Passive Earth Pressure

The maximum value 1 is reached when theMohr circle through the point representing the fixed value 3 touches the failure envelope for the soil. In this case, the horizontal stress is defined as the passive pressure ( pp) representing the maximum inherent resistance of the soil to lateral compression.

ppp KczKp 2

ppp KchKhE 2)2/1( 2

The force due to the passive pressure distribution is referred to as the total passive resistance (Pp). For a vertical wall surface of height H:


Example 1Lateral Earth Pressure

Example 2

h=5m

1=17kN/m3

c1=0

1=34o

2=19kN/m3

c2=10kPa

2=16o

h 1 =

2mh 2

=3m

A

B

C


A

B

C

h=5m

h 1=

2mh 2

=3m

011 aaA zKp kPaKhp aaB 4.10111 ＝上

kPaKcKhp aaaB 2.42 22211 ＝－下

kPaKcKhhp aaaC 6.362)( 2222211

mkNEa /6.712/3)6.362.4(2/24.10 ＝

10.4kPa4.2kPa

36.6kPa

Solution:Lateral Earth Pressure

6.3 Coulomb Theory of Earth Pressure

Coulomb’s theory (1776) involves consideration of the stability, as a whole, of the wedge of soil between a retaining wall and a trial failure plane. The force between the wedge and the wall surface is determined by considering the equilibrium of forces acting on the wedge when it is on the point of sliding either up or down the failure plane, i.e. when the wedge is in a condition of limiting equilibrium.

αβ

δ

G

h

A

C

B E R

Active earth pressure:

h

hK

a


α=10oβ=15o

δ=20o

4.5

m

A

B

α=10o

Ea

h/3

In terms of α=10o ， β=15o ， =30o ， δ=20o

480.0aK

mkNKhE aa /1.852

1 2 ＝

Solution:

The point of application of the total active thrust is not given by the Coulomb theory but is assumed to act at a distance of 1/3 H above the base of the wall.


6.4 the comparison of Rankine Theory and Coulomb Theory

1 analysis theory

Rankine theory Coulomb theoryLimit stress method

Slide wedge method

Solve the Soil pressure intensity p at

first

Solve the total earth pressure E at first

αβ

δE

Limit equilibrium condition


3 calculating error ——Rankine theory

WR

Ea'

Ea

δ0

WR

Ep

Ep'

Active pressureLarge

Passive pressuresmall


2 calculating error ——Rankine theory



the actual slip surface may not be a plane

Sliding surface


3 calculating error ——Coulumb theory

Sliding surface



2 calculating error ——Coulumb theory

Active pressure coefficient Ka (α=β=0)

δ=0 δ= /2 δ=

20 40 20 40 20 40

Sokolovsky

0.49 0.22 0.45 0.20 0.44 0.22

Rankine 0.49 0.217 0.49 0.217 0.49 0.217

Conlomb 0.49 0.22 0.45 0.20 0.43 0.21


3 calculating error ——compare with the exact value2 calculating error ——compare with the exact value

Passive pressure coefficient Kp (α=β=0)

δ=0 δ= /2 δ=

20 40 20 40 20 40

Sokolovsky 2.04 4.60 2.55 9.69 3.04 18.2

Rankine 2.04 4.60 2.04 4.60 2.04 4.60

Conlomb 2.04 4.60 2.63 11.8 3.52 92.6


2 calculating error ——compare with the exact value

1. Load on filling2. Stratified filling3. Water in filling

4 several kinds of active pressure


1 Rankin theory

1=z+q

pa= 3=qKa+zKa

HZ

1

3

qKaHKa

zKa

1. Load on filling


1. Load on filling


2. Coulomb theory

21 cos

2 cosa a aE H K qHKα

α β


1. Load on filling

3. Local loads—Rankin

theory


1. Load on filling

C

B

A

1 1 c1

2 2 c2 H2

H1

Rankin theory

)2

45(2)2

45(2 ooha tgcztgp


2. Stratified filling

earth pressure Pa=Kaszwater pressure pu=u

3. Water in filling


Impermeable foundation

朗肯理论

KaH2

KaH1

H2

水压力土压力


3. Water in filling

H2

H1

δ

Ea pw

Earth pressure Water pressure

Ea

pw

Coulombs theory


Impermeable foundation

3. Water in filling

Permeability coefficient of the filling is much smaller than of the foundation

H2

H1

saH2

H1

Effective stress

=total stress

u=0

Pore water

pressure

KasaH2

KaH1

Earht

pressure

透水地基


3. Water in filling

The flow net is needed to calculate the pore water pressure


3. Water in filling

Permeability coefficient of the filling is much smaller than of the foundation

6.1 general

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