9.4 Slope failure mechanisms in cohesive soils The most usual

Slopefailure mechanismsin cohesiuesoils 361
F =
c'+ (Tz - y*h) cos2I tan Q,,
yz sin B cosB
1 2+ ( 2 0x 4 - 9 . 8 1x 4 ) c o s 21 4 .x t a n2 4 o
-_',- : .!.!
2 0 x 4 x s i n 1 4 " x c o sl 4 o
9.4 Slope failure mechanisms in cohesive soils
Themost usualmethodsof providingan analysisof stabilityin slopesin
cohesive
soilsare basedon a considerationof rimitingplastic equilibiium.Fundamentally,
a_.conditionof limiting plastic equilibrium existsfrom the moment that
a shear
slipmovementcommencesand strain continuesat constantstress.It is first
necessaryto definethe geometryof the slip surface;the massof soil about to
move over
thissurfaceis then consideredasa freebody in equilibrium.The forcesor
moments
acting on this free body are evaluatedand those shearforces acting along
the
slip surfacecomparedwith the available shearresistanceoffered by the
soil.
severalforms of slip surfacemay be consideredfor cohesivesoilsas shown
_.
in
9
by cullmann in rg66, consistsof an
l'ig 6 The simplestof these,suggested
infinitelylong planepassingthrough the toe of the slope.Although
the analysis
of-thefree body equilibriumis simplein this case,the method y[lds factors
of
safetvwhich grosslyoverestimatethe true stability conditions.On the
other hand,
while the choice of a more complex surface,such as a log-spiral or
an irregular
shape'may produce resultsnear to the actual value, the analysistends
to be
long and tedious.For most purposes,a cyrindricarsurface,i.e. iircurar in
crosssection, will yield satisfactorily accurate results without involving
analytical
routinesof any great complexity.
The stability of a cut or built slopedependsvery largely on changesin
the
pore pressureregime.During the constructionof embankments
pore pressures
/+
+
Y
(al
/
(cl
Fig. 9.6 Types of failure surface
Drained stability - effectiuestressanalyses 379
2 . 0m
5 . 0m
Fig.9.22
Upperlayer: ci = 25 kPa Qi = 12" Tt = 18.0kNlm3
"tz= 19.5kNlm3
Lower layer: ci = 7 kPa
Q'z= 25"
The slip mass is divided into a convenient number of slicesof width D as in worked
example9.9. The averageheight of eachsliceis measuredoff a scaledrawing of the crosssectionin two componentsft, and hr. The weight of each sliceis therefore:
W=(Trhr+yrhr)b
As an alternativemethod to that usedin worked example9.9 (i.e. drawing the trianglesof
forces):in this examplethe angle a has beenevaluatedfor each slice:
a = arcsin(x/R)
From eqn [9.15]:
, _ 2 c ' l + L a n Q 'L [ ] / ( c o so - r u s e c a ) f
2 W s i na
which, sincethere are two layers,becomes:
A
. f
I
fciLo, + clLool+ | tan 4i) W(cosa - t seca) + tan Q'rZW@"ta - 6 seca)'l |
B
L
*.
F =
2W sina
378 Stability of slopes
7.0m
\.
groundwater
level
\{-
I
equapotential
l-
I
I
I
I
I
I
I
I
not to same scale
as others
Fig. 9.21
L e s = o R = 9 1 . t4#7t 1 8 . 5 8 =
2 9 .m
7
From eqn [6.16]
. _ c'Lor + tan Q' >N'
2r
_ l0 x 29.7 + tan 28" x 1052 = 1 . 0 5
814
A similar analysiscarried out on a microcomputer gaveF = 1.074.Another computer
analysisusing Bishop's simplified method gayeF = t . 2 2 .
vl/orkedexample9.70 Determinethefactor of safety in terms of effectiue stressof the
slopeshownin Fig. 9.22 in respectof the trial circle shown.Assumethat thepore pressure
ratio r,= 0.3 and that the soilpropertiesare asfollows:
Drained stability - effectiue stressanalyses
3i.7
y l c ' b +(W-u b )ta n Q'l se ca
r= *J.
[e.18]
2 w s r n a ,l+tan utang,lF
The procedureis commencedby assuminga triar varue for
the ,F on the righthand side and then, using an iterative p-""rr, to converge
on the true value
of ,F for a given trial circle. This is the routine procedurf
commonly used in
programsdesignedfor use on computers.Many of th. program
packagesnow
availableoffer also the inclusion of multiJayer conditions,s,ircharge
loads,variablepore pressuredistribution and eventhe provision of berms
and drains.In
most problems, it is acceptableto adopt a constant average
varue for ru. The
factors of safety computed by this -"ihod may be slight
underestimates,but
with errors not usually exceeding3 per cent, exceptin occasional
unusual cases
with deepbasefailure circlesunJ F t"r, than unity. More accurate
lower factors
of safety are claimed for methods which account for the variation
in seepage
forceson and in the slice(King, l9g9). However,suchrefinements
dependeven
more on good estimatesof pore pressures.Morrison and
Greenwood(19g9)
havefurther examinedthe assumpiionsmade in this method.
vltorkedexample9.9 Determinethe
factor of safety in terms of effectiue stress
Jbr the
slopein Fig. 9.21 inrespect of the triar circreihown. The soirpropeiti^
o* as fbilows:
c' = I0 kPa Q' = 28. T = Ig kNlm3
Thepore pressuredistributionalong the trial circle is obtained
by sketchingequipotentials
at eqchslice centre.
Figure 9.21 showshow a partialy graphical sorution may
be obtained. The slip mass
has beendivided into slicesofwidth 3 m and the sectiondrawn
to an approp'ate scale.
The averageheight of each slice(ft) is scaledoff the diagram
and its w#ht calculated.
14t= yhb = 18 x h x 3 = 54.0/zkN/m
ofthe chord at the baseofeach sliceis scaledoffand the pore pressure
fbrce
lT ]:lttn
calculatecl.
u l = h * x 9 . 8 1x /
A triangle of forcesis drawn at the baseof each sliceto obtain
varuesof,A/ and z.
The calculationsare tabulated below:
Slice h (m)
no.
W
(kNlm)
I
2
0.6
)z
J
4
5
6
7
+.o
6.4
7.4
8.3
7.2
8*!
J . l
130
248
346
400
448
389
246
'r^
* Width = 3.7 m
h, (m)
0.6
1.7
3.6
4.6
5.3
5.1
4.1
1.3
1(m)
).2
3.1
3.0
J,I
3-Z
3.5
3.9
6.9
N
ul
3t
t9
r28
s2
248 106
341 140
378 166
381 175
298 t57
150 88
N'
12
76
r42
201
2r2
206
141
62
> 1052
_8
-20
0
57
130
2t8
251
186
814
376 Stabilityof slopes
value for F which may be as much as 50 per cent on the low side. Errors may
also arisewhen ru is high and the circle is deep-seated
or has a relativelyshort
radius.In suchcases,Bishop'smethodis preferable.
Bishop'ssimplifiedmethod
In reasonablyuniform conditions and when also r, is nearly constant,it may be
assumedthat the tangentialintersliceforcesare equaland opposite,i.e. xr=f,,
but that Er+ Er(Fig. 9.20).
For equilibriumalong the baseof the slice:
c'L + N' tanQ'
o =w sina - lt :w sina F
F
L(c'l + N' tan Q')
F -
So that
2 W s i na
For equilibrium in a vertical direction:
0 = W - - l y ' ' c ous - u l c o s a - l L l , r n o
F
TL^_
I nah
=w -N' cosa - ulcosd - lt
F
^r, W (c'lF)l sin g - ul cosa
/\/
N' tunQ'
rino rinu
F
-
c o sd + ( t a n Q ' l F ) s i n a
After substituting for /:
b sec a and N' rearranging
Wsino
(
\
\
Fig. 9.20 Bishop's simplified slice
Drainedstability- effectiuestress
analyses 375
E, andE'z= normal intersliceforces
X, andXz = tangentialintersliceforces
Theeffectsof any surchargeon the surfacemust be includedin the computation
ofthe body weight and other forces.
At the point of limiting equilibrium, the total disturbing moment will be
exactlybalancedby the moment of the total mobilisedshearforce alons AB.
r . / R = H) F* l o = Z r y s i na R
-) ' "
giving
D
a
-
l
I r^/
-
2W sina
Now in terms of effectivestress,
and
xf = c'+ o'"tan f
Trl= c'l + N' Ian f
, _ I c'l+2 N'tanQ'
2 W s i nu
Sothat
or if the soil is homogeneous
, _ c'Lou + lan Q' L N'
2W sina
where Lo.n= a.rclength AB = 9R
[e.16]
A lot dependson how the valuesof N'are obtained.A numberof methodshave
beensuggested,somerelatively simple and somewhich are quite rigorous. The
mostaccurateestimatesmay be expectedfrom rigorousmethods,but may only be
possibleif a computer routine can be employed.A compromisemay be arrived
at by combining a simpler method of analysiswith an increasedfactor of safety.
Felleniuso
method
In this method, it is assumedthat the interslice forces are equal and opposite
and canceleach other out, i.e. Er = Ez and X, = Xz. It is now only necessaryto
resolvethe forces acting on the baseof the slice,so that:
N'=Wcosu-ul
= yhb cos a* ub seca
(l = b secu)
orputting u=ruyh
1tJ'= yh(cos a - r" seca)b
o r I N ' = T b D h ( c o sa - r u s e c a )
Then substitutingin eqn [9.13]:
, _ !'Lou + yb tan Q' L h(cosa. - ru seca,)
LW sina
le.r7l
The number of slicestaken should not be lessthan five, and obviously a larger
number would yield a better estimateof ,F.Even so, this method tends to sive a
374 Stabilily of slopes
(b) When D = 1.5,B = 35oand 0, = 0, then from Fig. 9.18(b):
N=0.168 and n=0.6
l-actor of' safety,
-
r =
40
= -1.65
0.168x18x8
Thus, the presenceof the harder layer constrains the failure mode to a smaller
critical circle and so the factor of safetyis larger, and the break-out point (n11)of
this circlewill be 0.6 x 8 = 4.8 m.
9.10 Drained stability - effective sfress analyses
Stability analysesshould be carried out in terms of effectivestressesin problems
where changesin pore pressuretake place, such as existing embankmentsand
spoil tips; also to estimatethe long-termstability of slopesand in the caseof
overconsolidated
claysfor both immediateand long term conditions.Because
of
the variationsin the stresses
alonga trial slip surface,the slip massis considered
as a seriesof slices.A trial slip circle is selectedhaving a centre o and a radius
R (Fig. 9.19),and the horizontaldistancebetweenthe two endsA and B divided
into slicesof equalbreadthb.
The forces acting on a slice of length I m will be as follows:
W = the body weight o, tA" tfiss = yhb
N' = the effectivenormal reacting force at the baseof the slice
T = the shearingforce induced along the base
=Wsinu
R, and Rz = forces imposed on the sidesfrom adjacent slices- which may be
resolvedinto:
e
___.,-{L
base assumed
to be straight
of length /
{a}
Fig. 9.19
(bl
Method of slices
(a) Division of slip mass (b) Forceson a slice
Taylor'sstability numbermethod 373
v-zc
o.25
o"ls
a 0.20 D = *
o
E
.=
0.15
0.10
o
a
0.05
0.181
t4
--';
e{,'i;,
tr
'ili:
tr
u!,'
10
20
==:
=?-
53"
30
40
50
60
Slopeangle,0 (deg)
70
80
90
(b)
F i g .9 . 1 8 C o n t i n u e d
the critical circle may pass in front of the toe and the chart shown in Fig. 9.18(a)
is used. When the critical circle will be restricted to passing through the toe, the
heavy broken lines on the chart must be used. The value of n, giving the breakout point of the critical circle in front of the toe, can be obtained from the light
broken lines.
Workedexample9.7 A cutting in a saturatedclay hasa depthof 10 m. At a depthof 6 m
belowthefloor of the cutting there is a layer of hard rock. The clay has an undrained
cohesion
of 34 kPa and a bulk unit weightof 19 kNlm3. Calculatethe maximumsafeslope
that will prouidea factor of safety of 1.25 againstshort-termshearfailure.
Referto Fig. 9.18(a).
1 1= l 0 m a n d D H = 1 6 m
.'. D=1.5
Jl= --4= 0.143
Requiredstability number,' 1,' : -l . 2 5 y H 1 . 2 5x 1 9x 1 0
Thepoint on the chart locatedby D = 1.5 and N= 0.143givesa slopeangleof B = 13".
Also. from the chart. n = 0.2. Hence the circle will break out 2.0 m in front of the toe.
Workedexample9.8 A cutting in a cohesiuesoil has a slopeangle of 35" and a uertical
heightof 8 m. Using Taylor's stability method,determinethefactor of safetyagainstshear
failurefor thefollowing cases:
(a) c,= 40 kPa "l = 18 kNlm3 D is large
rb) c,= 40 kPa "t = 18 kNlms D = .1.5
(a) When D is large,with B < 53' then N = 0.181
j9= r.S:
Factor of safety, P = ---0.l8lx18x8
372 Stability of slopes
o.te!
0.
t >
--rt =
"llf
.ll
= 0 .1 4
o
-o
E
t
c 0 .1 3
4
depth factor; D
.=
lt
o
o
-
nH
l+--l
CaseA: usefull linesfor N
and short-dashlines
tor n
B: toe circles only.
Use long-dash lines for N
1
(al
2
Fig.9.18 Stabilitynumbersfor total stressanalyses
(a) In terms of depth factorD (bl Toe circlesin terms of slopeangleB
Hence.
or since
Ir -
r
-u
NvH
te.l5l
= c,/F, required c-ou.= NyH
cmob.
Values of N related to the slope angle B, the angle of shearing resistance @"and
the depth factor D are given in the charts shown in Figs 9.18(a) and (b). For
slope angles greater than 53o, the critical circle passesthrough the toe of the
slope and the chart shown in Fig.9.l8(b) is used. For slope angles less than 53o,
Taylor's stability number method
108654 32.5 2
1.5
0.75
371
0.5
\
\
\
\
Y"
H
N
.
0.5
0
\
1
0
2
0
3
0
4
slopeanglep"
0
50
60
Fig.9.17 First trial location of the critical circle
9.9 Taylor's stability number method
In 1948,D. w. Taylor proposeda simpremethod of determiningthe
minimum
factor of safetyfor a sJopein a homogineoussoil. Using a totaistress
analysis
and ignoring the possibitityof tensioncracks,he produceda series
of curves
which relate a stability number(N) to the slope angle
B.
consider the basicexpressionusedin a total stressanalvsis
c,,RL
n = -Wd 0rom eqn [9.12])
Itwill be seenthat Ln H and Wn
TH2,i.e.L= K1H, W= KrTH,
cuRHKr
Then p YH2Krd
The stability numberis dependenton the geometryof the slip circle and
may be
definedas:
"u
N=Krd =
K,R
FyH
Undrained stability - total stressanalyses 365
R
\\-
f" = 9.58 m
\"
= 4.32m
Fis. 9.10
(b) The effect of the tension crack is to reducethe arc length from
AB to AC.
Depth of tensioncrack,
Sectorangle,
Area of slip mass,
Centroiddistancefrom O,
In this case,
Then from eqn [9.13]:
zo= 2 x 40/19.5= 4.32m
0.= 67.44o
A = 11.64m2
d = 5.g6m2
p_ = 0
p _ cuR,O"
Wd
_ 40 x 17.432x 67.44x n
71.64x18.5x5.86x180
= 1 4 3 0 4= ,: o o
7766
(c) When the tensioncrack is full of water, a horizontal force p,
will be exertedon the
slip mass.
P * = I y * z ' o = j x 9 . 8 1x 4 . 3 2 2 =9 1 . 5 4k N / m
The leverarm of P, about O, !" = 6.7 + 2 x 4.3213= 9.5gm
Then from eqn [9.13]:
r- =
c.,R20.
Wd + P*y.
14 304
= 1.65
1766+91.54x9.58
364 Stability of slopes
T
I
lr .
t
'c
^/
2c..
f
fC"
c
hydrostatic pressure
in tension crack
Fig. 9.9
Effect of tension crack in total stress analysis
slip circle arc and the tension crack. No shearstrength can be developedin the
tension crack, but, if it can fill with water, allowance must be made for the
hydrostatic force P*, which acts horizontally adding to the disturbing moment:
P* = iY*rZ
Taking this into account,togetherwith the factthat the slip circle arc is reduced,
the factor of safetyexpressionbecomes:
f
n
crR20"
[e.13]
= -
Wd + P*y"
Worked example9.4 A cutting in a saturatedclay is inclinedat a slopeof I uertical:L5
horizontaland hasa uerticalheightof 10.0m. Thebulk unit weightof the soil is 18.5kNlm]
and its undrainedcohesioni.s40 kPa. Determinethefactors of safety against immediate
shearfailure along the slip circle shownin Fig. 9.10: (a) ignoring the tensioncrack, (b)
allowingfor the tensioncrack empty of water, and ( c) allowingfor the tensioncrack when
full of water.
The factors of safetyagainstimmediateshearfailure may be obtained using the total
stressmethod of analysis.Firstly, it is necessaryto establishthe geometryand areaof the
slip mass.
(a) In the case ignoring the tension crack, the slip mass is bounded by the ground
surfaceand the circular arc AB, for which the following may be calculated.
Radius,R= OA={(5,+ 16.7\=t7.43m
Sectorangle,
0= 84.06'
Area of slip mass,
A = 102.1m2
d = 6.54m
Centroiddistancefrom O,
T h e nf r o m e q n [ 9 . 1 '2 ] : Ow= ' i r \ .d" t
4 0 x 1 7 . 4 3 x28 4 . 0 6 x n
lorJrt8ir6s+*ru=t-44
Undrained stability - total stress analyses
363
Fis.
s.s rotat
*.. ,lil-r.,"
Figure9.8 showsthe cross-section
of a slopetogetherwith a trial slip circleof
radiusR and centreo. Instability tendsto be causeddue to the moment of the
bodyweight W of the portion abovethe slip circle.
Disturbing moment = Wd
Thetendencyto move is resistedby the momentof the mobilisedshearstrength
actingalong the circular arc AB.
Lengthof arc
AB = R0
Shearresistanceforce along AB = c.Rg
Shearresistance
moment
Then factor of safety,
= c..R20
shear resistance moment
F -
disturbing moment
cuR20
Wd
[e.r2]
The values of w and d arc obtainedby dividing the shadedarea into slicesor
triangular/rectangularsegmentsand then taking area-momentsabout a vertical
axispassingthrough the toe, or other convenientpoint.
Tensioncracks
In cohesivesoils, a tensioncrack tends to form near the top of the slope as
the condition of limiting equilibrium (and failure) develops.From Chapter 8
[eqn8.13]it will be seenthat the tensioncrack depth may be taken as
g^=12
v
The development of the slip circle is terminated at the tension crack depth
and so its arc length is really AC as shown in Fig. 9.9. The free body weight W
of the slipping mass is the shaded area bounded by the ground surface, the
362 Stabilityof slopes
will rise and, after construction, they will gradually fall. In cuttings, however,
excavationcausesan initial fall in pore pressures,but as seepagedevelopsthey
gradually rise. Effective stressesand therefore shear strengthsare generallyin'
verselyrelated to pore pressures.The most critical (lowest)factor of safetymay
therefore be expectedto occur immediately after or during the constructionof
an embankment;after this the soil will gradually get stronger. In contrast,the
shearstrengthin a cutting diminisheswith time and so doesthe factor of safety.
Thus, it is necessaryto consider both short-term (end-of-construction)and
long-term stability. In this context it is convenientto think of short-term conditions as being completely undrained,in which the shear strength is given by
t = cu.ln the next section,the first caseto be consideredwill be that of undrained
stability of a slopein a saturatedclay; this type of analysisis often referredto as
a total stress,method.
For long-term problems,and problems where changesin conditionsmay
occur long after the end of construction (suchas the suddendraw-down of level
in a reservoir),a form of effectiueslressanalysisis required.Thesemethodsmay
take the form of either a force- or moment-equilibriumanalysis,involving plane,
circular or irregular slip surfaces.For complex problems, stresspath and slip
line field methodsare used.The shearstrengthparametersmust be chosenwith
care (seeSection9.2).
9.5
Undrained stability - totalsfress analYses
A total stressanalysis may be applied to the case of a newly cut or newly
constructedslope in a fully saturatedclay, the undrained shear strength being
T = cu.It is assumedthat the failure surfacewill take the cross-sectionalform
of a circular arc, usually referred to as the slip circle. The centre of the critical
slip circle will be somewhereabove the top of the slope. The critical (failure)
slip circle is one of an infinite number of potential circles that may be drawn
having different radii and centres(Fig. 9.7). Somecircleswill passthrough the
toe of the slope and somewill cut the ground surfacein front of the toe'
The critical circle is the one along which failure is most likely to occur and for
which the factor of safetyis the lowest. A number of trial circlesare chosenand
the analysisrepeatedfor eachuntil the lowest factor of safetyis obtained.
Fig. 9.7 Slip circles at different radii and centres
380 Stability of slopes
The resultsare tabulated below for a solution in which ten sliceswere taken:
Slice
no-
I
2
3
4
5
t')
7
8
9
h, (m)
0.00
0.00
0.00
0.55
2.00
3.30
4.70
5.00
5.00
h, (m)
0.6
2.1
3.9
5.0
4.8
4.4
3.5
2.5
0.7
W
(kNtm)
x (m)
23.4
8t.9
152.1
214.8
259.2
290.4
305.7
277.5
207.3
5.67
4.00
2.00
0.00
2.00
4.00
6.00
8.00
10.00
u (de7.)
cos cx-
24.10
16.70
8.28
0.00
8.28
16.70
25.60
35.20
46.00
0.584
0.64s
0.686
0.700
0.686
0.64s
0.569
0.450
0.26s
2.9
0.0
125.3
T
11.8
58.1
t3.7
52.8
104.3
1s0.4
t77.8
187.3
173.9
124.9
s4.5
-9.6
-23.s
21.9
0.0
31.3
83.5
132.1
160.0
149.0
1040
2N'(A-+F)
10*
N'
fu sec a
-0.039
0t
2 : f ( A - + B)
106.4
613
* Width = 2.4 m
t When cos a - ru seca < 0, N' is set to 0, sinceN' cannot be negative
N' = W(cos a-
rusec a)
Lo.=go.R =79.9x
ft
180
T = W sin cx
x13.39=19.37m
Lrs= 23.7*
#'
= 5.75m
* 13.89
x 5.75+ 7 x 19.39]+ [tanl2"(0)+ tan 25"(1040)]
Then " _ [25
613
_
279
+ 485
------..-:blJ
_ !:=!
A similar analysis carried out on a microcomputer gave F : l.2j . Another computer analysis using Bishop's simplified method gave F = 1.42.
9.11 Effective sfress stability coefficients
A method involving the use of stability coefficientssimilar to that devisedby
Taylor, but in termsof effectivestress,was suggestedby Bishop and Morgenstern
(1960).The factor of safety(,F) is dependenton five problem variables:
(a)
(b)
(c)
(d)
(e)
slopeangleB
depth factor D (as in Taylor's method- Fig. 9.18)
angleof shearingresistance@'
a non-dimensionalparameterc'lyH
pore pressurecoefficientr"
The factor of safetyvaries linearly with r, and is given by
F:n1-flru
[e.re]