bra是什么边坡稳定经典英文教材及翻译之原文

2021年1月26日发(作者：伯蒂)

CHAPTER 9
Stability of Slopes


9.1

Introduction

Gravitational and seepage forces tend to cause instability in natural slopes,
in
slopes formed by excavation and in the slopes of embankments and earth dams.
The
most
important
types
of
slope
failure
are
illustrated
in

rotational
slips
the
shape
of
the
failure
surface
in
section
may
be
a
circular
arc
or
a
non-circular curve
．
In general
，
circular slips are associated with homogeneous soil
conditions and non-circular slips with non-homogeneous conditions
．
Translational

and compound slips occur where the form of the failure surface is influenced by















the presence of an adjacent stratum of significantlydifferent strength
．

Translational
slips
tend
to
occur
where
the
adjacent
stratum
is
at
a
relatively
shallow depth below the surface of the slope:the failure surface tends to be plane
and
roughly
parallel
to
the
nd
slips
usually
occur
where
the
adjacent stratum is at greater depth
，
the failure surface consisting of curved and
plane sections
．


In
practice,
limiting
equilibrium
methods
are
used
in
the
analysis
of
slope
stability.

It
is
considered
that
failure
is
on
the
point
of
occurring
along
an
assumed or a known
failure surface
．
The shear
strength required to
maintain a
condition of limiting equilibrium is compared with the available shear strength of
the soil
，
giving the average factor of safety along the failure surface
．
The problem
is considered in two dimensions
，
conditions of plane strain being assumed
．
It has
been
shown
that
a
two-dimensional
analysis
gives
a
conservative
result
for
a
failure on a three- dimensional(dish-shaped) surface
．


9.2


Analysis for the Case of
φ
u
=0

This analysis, in terms of total stress
，
covers the case of a fully saturated clay
under
undrained
conditions,
i.e.
For
the
condition
immediately
after
construction
．
Only moment equilibrium is considered in the analysis
．
In section,

the
potential
failure
surface
is
assumed
to
be
a
circular
arc.
A
trial
failure
surface(centre O
，
radius r and length L
a
)is shown in Fig.9.2. Potential instability is
due
to
the
total
weight
of
the
soil
mass(W
per
unit
Length)
above
the
failure
surface
．
For equilibrium
the shear strength
which must be mobilized along the
failure surface is expressed as

where F is the factor of safety with respect to shear strength
．
Equatingmoments
about O
：



Therefore
























(9.1)

The moments of any additional forces must be taken into account
．
In the event
of a tension crack developing
，
as shown in Fig.9.2
，
the arc length L
a
is shortened
and a hydrostatic force will act normal to the crack if the crack fills with water
．
It
is necessary to analyze the slope for a number of trial failure surfaces in order that
the minimum factor of safety can be determined
．



Based on the principle of geometric similarity
，
Taylor[9.9]published stability
coefficients for the analysis of homogeneous slopes in terms of total stress
．
For a
slope of height H the stability coefficient (N
s
) for the failure surface along which
the factor of safety is a minimum is

























(9.2)
For
the
case
of
υ
u

=0
，
values
of
N
s

can
be
obtained
from

coefficient N
s
depends on the slope angle
β
and the depth factor D
，
whereDH is
the depth to a firm stratum
．

Gibson
and
Morgenstern
[9.3]
published
stability
coefficients
for
slopes
in
normally
consolidated
clays
in
which
the
undrained
strength
c
u
(
υ
u

=0)
varies
linearly with depth
．


Example 9.1

A 45
°
slope is excavated to a depth of 8 m in a deep layer of saturated clay of

unit weight 19 kN
／
m
3
:the relevant shear strength parameters are c
u
=65 kN
／
m
2
and
υ
u
=0
．
Determine the factor of safety for the trial failure surface specified in
Fig.9.4.
In Fig.9.4,the cross-sectional area ABCD is 70 m
2
.
Weight of soil mass=70
×
19=1330kN
／
m
The centroid of ABCD is 4.5 m from O
．
The angle AOC is 89.5
°
and radius OC
is 12.1 m
．
The arc length ABC is calculated as 18.9m
．
The factor of safety is
given by
：




This
is
the
factor
of
safety
for
the
trial
failure
surface
selected
and
is
not
necessarily the minimum factor of safety
．

The minimum factor of safety can be estimated by using Equation 9.2.
From Fig.9.3
，
β
=45
°
and assuming that D is large
，
the value of N
s
is

9.3


The Method of Slices

In this method the potential failure surface
，
in section
，
is again assumed to be a
circular arc with centre O and radius r
．
The soil mass (ABCD) above a trial failure
surface(AC)
is
divided
by
vertical
planes
into
a
series
of
slices
of
width
b,
as
shown in base of each slice is assumed to be a straight line
．
For any
slice the inclination of the base to the horizontal is
α
and the height, measured on
the
centre-1ine,is
h.
The
factor
of
safety
is
defined
as
the
ratio
of
the
available
shear strength(
τ
f
)to the shear strength(
τ
m
) which must be mobilized to maintain
a condition of limitingequilibrium, i.e.


The factor of safety is
taken to
be the same for each slice
，
implying that there
must be mutual support between slices
，
i.e. forces must act between the slices
．

The forces(per unit dimension normal to the section)acting on a slice are
：

total weight of the slice
，
W=
γ
b h (
γ
sat
where appropriate)
．

total normal force on the base
，
N (equal to
σ
l)
．
In general this
force has two components
，
the effective normal force N
'
(equal to
σ
'
l ) and
the boundary water force U(equal to ul )
，
where u is the pore water pressure
at the centre of the base and l is the length of the base
．

shear force on the base
，
T=
τ
m
l
.

total normal forces on the sides,E
1
and E
2
.
shear forces on the sides
，
X
1
and X
2
.
Any external forces must also be included in the analysis
．



The
problem
is
statically
indeterminate
and
in
order
to
obtain
a
solution

assumptions must be made regarding the interslice forces E and X
：
the resulting
solution for factor of safety is not exact
．



Considering moments about O
，
the sum of the moments of the shear forces T
on
the
failure
arc
AC
must
equal
the
moment
of
the
weight
of
the
soil
mass
ABCD
．
For any slice the lever arm of W is rsin
α
,
therefore
∑
Tr=
∑
Wr sin
α

Now,

For an analysis in terms of effective stress
，


Or




















(9.3)
where
L
a

is
the
arc
length
AC
．
Equation
9.3
is
exact
but
approximations
are
introduced in determining the forces N
'
．
For a given failure arc the value of F will
depend on the way in which the forces N
'
are estimated
．


The Fellenius Solution

In
this
solution
it
is
assumed
that
for
each
slice
the
resultant
of
the
interslice
forces is zero
．
The solution involves resolving the forces on each slice normal to
the base
，
i.e.
N
'
=WCOS
α
-
ul

Hence the factor of safety in terms of effective stress (Equation 9.3) is given by














(9.4)
The components WCOS
α
and Wsin
α
can be determined graphically for each
slice
．
Alternatively
，
the value of
α
can be measured or calculated
．
Again
，
a series
of trial failure surfaces must be chosen in order to obtain the minimum factor of
safety
．
This solution underestimates the factor of safety
：
the error
，
compared with
more accurate methods of analysis
，
is usually within the range 5-2%.


For an analysis in terms of total stress the parameters C
u
and
υ
u
are used and
the value of u in Equation 9.4 is zero
．
If
υ
u
=0 ,the factor of safety is given by





























(9.5)