S. Sarma (1973) - Stability Analysis of Embankments and Slopes
S. Sarma (1973) - Stability Analysis of Embankments and Slopes
S. Sarma (1973) - Stability Analysis of Embankments and Slopes
and slopes
INTRODUCTION
The slip surfaces that may develop during failure in natural or man-made slopes are generally
non-circular.
Regardless of whether failure is caused by static or earthquake
forces, slip
surfaces will be of a general shape rather than strictly circular.
With increasing nonuniformity and degree of zoning of the slope material, departure from circularity of the failing surface increases.
Ambraseys (1959) has shown that even for homogeneous
elastic cases,
the potential
failure surface during earthquake
shaking is non-circular.
As earthquake
inertia forces become larger, or zoning of the slope material becomes stronger, so results from
circular slip analysis become more unrealistic.
The failure surface of the Lower San Fernando
dam, produced by the San Fernando Earthquake of 9 February, 1971, is indeed non-circular
(Jennings, 1971). Case histories studied by Morgenstern and Price (1965) show that even
without earthquake loading, there are cases of non-circular slip surfaces both in natural slopes
and in embankments.
So far, stability analysis methods based on general slip surfaces aim at the determination
of the minimum factor of safety of the slope. These methods are based on the principle of
limiting equilibrium and the method of slices. Kenney (1956), Janbu (1957) and Nonveiller
(1965) developed techniques of analyses based on these methods which, however, do not
satisfy all conditions of equilibrium. It is obvious, therefore, that these methods are in error
* Department of Civil Engineering, Imperial College of Science and Technology,
London.
424
S. K. SARMA
by a certain unknown amount. The degree of inaccuracy involved in these methods depends
on the extent of inaccuracy associated with the assumptions made. The first known methods
which satisfy all conditions of equilibrium are those of Bishop (1955) for circular arc slip
surfaces and of Morgenstern and Price (1965) for general slip surfaces. The MorgensternPrice method involves many iterations and cannot be used without the aid of a computer.
Bell (1968) produced another statically accurate solution which also cannot be applied without
a computer.
Madej (1971) modified Janbus method to satisfy equilibrium but needs 20-25
iterations to determine a factor of safety. Therefore, there is still need of a simple method of
analysis of a general slip surface which is accurate in the sense that it satisfies all the conditions
of equilibrium.
The method proposed in this Paper is developed as a means of computing the critical horizontal acceleration that is required to bring the mass of soil bounded by the slip surface and
the free surface to a state of limiting equilibrium. This critical acceleration is therefore a
measure of the static factor of safety. To determine the critical acceleration, no iteration is
necessary. The usual factor of safety is obtained on a simple iterative basis which requires
only about three iterations and does not contain any problem of convergence. If, however,
the critical acceleration is to be determined by the other methods mentioned already, the amount
of computational work involved becomes too large. The simplicity of the proposed
stability analysis method will become apparent from the formulation of the problem.
E=E;
Pw
N&U
E, and E,+%
i+
if I
Fig. 1
STABILITY
ANALYSIS
OF EMBANKMENTS
AND
42.5
SLOPES
The advantage of the method is that the aid of a computer is not essential and the problem
of non-convergence
does not exist. The accuracy associated with the Morgenstern-Price
method is not lost in the simplicity of the present method.
However, in the present solution,
as in any stability analysis solution, the physical acceptability of the solution must be checked
before the result itself is accepted, e.g. internal forces obtained from the solution must not
violate failure criteria and tension must not be implied within the soil mass; see, for instance,
Whitman and Bailey (1967).
FORMULATION
x:&=0
xOXi=O
From the vertical
and horizontal
equilibrium
(la)
.*.......(lb)
D&
may be obtained
.
(2)
(3)
It is assumed that, under the action of the force KW, the complete shear strength of the surface
is mobilized.
This means that the factor of safety is equal to one and therefore K represents
the critical acceleration Kc as a fraction of gravity.
Then
Ti = (Ni -
(4)
(5)
where
Vi = R,iWiseca,
and where Rui is the pore pressure ratio as defined by Bishop and Morgenstern
From equations (2), (3), (4), (5)
Ni = [-c&
(1960).
tan cc1+ Wi(l+Rui tan cl,tan &) cos +',/cos (+'i-ai)]- DX, cos $'JCOS ($'i-cq)
(+'i-aJ
- DX,
= Dt-KWI
(6)
(7)
(8)
(9)
where
Di = Wi tan (+i - ai) + (C',biCOS$'i- R,i Wi sin $i) set ai/COS (+'i- ai)
Considering
the equilibrium
KW, = 2 Di
(10)
(11)
426
S. K. SARMA
(14)
(15)
(4t-d+(xi-xhl
= CD*
= C W,h-x~)+C Dt(yi-Yg)
* (16)
Equation (13) suggests that the shape of distribution of the X forces, but not their magnitude,
is known. The magnitudes will be known from the solution of equation (16). Since F, is
assumed to be known, equations (15) and (16) can be solved simultaneously to obtain h and K.
A=
s,/s,
K = (A,-A&)/C
W,
........
(17)
........
(18)
where
S1 = 2 Di
.
.
.
.
. . (20)
. . (21)
& = .Xfh-~~)tan
(di-4+(~t-d1
S, = 2 F, tan (#J~
-CQ)
(19)
(22)
The value of K gives the critical acceleration Kc for the surface as a fraction of gravity, AF,
gives the change of X force across the ith slice. From equation (8) the change of E force
across the slice is obtained.
DXi = XF,
DE, = Di-KW,-DX,tan@,-CC,)
(23)
From the known initial condition at point A where X, = El = 0 all X and E forces can now be
determined. These E forces will satisfy equation (la). The factor of safety on the vertical
sections of the slices (i.e. the local factor of safety) can now be calculated,
FL1 = [(Ei -P,J
(24)
Where tan rJI and Et are the weighted average shear strength parameters of the section i.
Taking moment equilibrium of the ith slice alone, the point of application of the E forces
can be determined. Taking moment about the centre of the base of the slice one obtains
~(~1 = [E,z,-O*5bi tan cc,(E,+E,+,)-0.5b,(X,+Xi+,)]/E,.,
Delivered by ICEVirtualLibrary.com to:
IP: 86.0.218.4
On: Fri, 08 Oct 2010 12:58:16
(25)
SWBLLITY
ANALYSIS
OF EMBANKMENTS
AND
427
SLOPES
Again starting from known initial condition z1 = 0 all points of application can be determined.
The accuracy of the proposed method was checked against that of the Morgenstern-Price
method by the following procedure.
For a particular surface, the earthquake acceleration
used in the Morgenstern-Price solution was increased progressively until the factor of safety
was reduced to one. The X values that were obtained from this solution were used to determine the Fi values which were subsequently used in equations (21) and (22), to determine the
values of X and K. The results obtained differed only in the fourth significant figure.
It is now apparent that the problem has an infinite number of solutions which depend on
the infinite variation of Ft. This is a situation similar to that arising in the MorgensternPrice solution. Of this infinite number, only a few solutions are acceptable; the rest must
be rejected on the grounds of physical reasoning. As argued by Morgenstern and Price
(1965) and Whitman and Bailey (1967), only those solutions are acceptable which do not
violate the failure criterion of the soil mass above the slip surface and which at the same time
do not cause tension in the material. The solution is acceptable if FL, is greater than unity
and 0 < z,/H, < 1 in all sections. It has been found that the following definition of Xi gives
satisfactory results. The derivation of the formula is given in the Appendix.
Xi = xh[Ki-R,,)yHi2
tan
$J~+c&,]
(26)
(27)
(28)
where
K, = 1 -sin &(l - 2RJ sin $1+ 4~~cos +JyHi)
1
1 + sin #J~sin pi
pi = 2ai-$,
fi is a number to be selected, usually 1. This represents the reciprocal of the expected local
factor of safety. However, the true local factor of safety is obtained as part of the solution
which is not necessarily the same as the expected one.
hF* = X,+,-X*
(29)
Formulae (26), (27), (28) are derived for the case in which the material and the pore pressures
are homogeneous along the vertical boundaries of the slices. Other formula: which take
into consideration the non-homogeneity along these boundaries are given in the Appendix.
In order to obtain the stutic factor of safety, the strength parameters of the material along
the slip surface must be reduced by a known factor of safety, and the critical acceleration
computed. The value of the factor which gives zero critical acceleration is the factor of safety,
which obtains without the earthquake forces. However, the critical acceleration itself may
be used as a measure of the static factor of safety and this iteration is then unnecessary.
APPLICATION
In almost all cases that are tried, J = 1 gives satisfactory results. In those cases, where X
and E forces violated the failure criterion, the value off, was changed so as to eliminate this
violation. It is observed that if the slip surface is a reasonable one, an acceptable solution is
obtained withJ;:= 1. If the failure criterion is violated at one or two of the vertical sections
only, this may be due to the discontinuity of the slip surface. In this case, a satisfactory
solution can be obtained by smoothing the surface or by reducing the value off for those
sections to less than unity. If the failure criterion is violated in a larger proportion of the
mass, no reasonable X forces can be easily found for that surface. However, this point needs
further investigation.
Delivered by ICEVirtualLibrary.com to:
IP: 86.0.218.4
On: Fri, 08 Oct 2010 12:58:16
S. K. SARMA
428
Tables l(a) and l(b) compare values obtained by the Morgenstern-Price method and the
proposed method using equation (26). The Kc values were obtained in the Morgenstern-Price
method by gradually increasing the earthquake acceleration coefficient until the factor of
safety was reduced to one. The corresponding static factor of safety obtained in this process
is shown.
In Table 2 results obtained for non-homogeneous cases are shown. The results of the
proposed method were obtained using equation (48). The factors of safety in the proposed
method were obtained by reducing the strength of the material on the slip surface until K,
became zero. The corresponding Kc value for the full strength is also shown.
Scale of Feet
0
do
Fig. 2
Table l(a).
No.
Table l(b).
No.
F.S.
KG
4.213
3.126
2,594
3.018
2.485
2.913
2.562
2.460
2.569
2.908
0.8064
0.676
0.529
0.678
0.539
0.635
0.578
0.533
0.542
0.561
R,=0.40
Authora
KC
0.8073
0.666
0.527
0.666
0.535
0.629
0.570
0.532
0.541
0.563
(Fig. 2)
Morgenstern-Price1
F.S.
KC
1 .7424
1.4494
1.7114
I.480
1.461
0.271*
0.1704
0.2484
0.1904
0.176
Authora
KC!
0.245
0.164
0.235
0.176
0.170
STABILITY
ANALYSIS
Table 2.
OF EMBANKMENTS
AND
Morgen$erSn-Price
_-
Author
. .
1.014
(see Fig. 3)
429
SLOPES
Non-homogeneous case
Note
F.S.
K,
0.98
1 .oo
- 0.005
0.000
-A=1
f, = varied
f (xA;t==oIorgenstern-Price
-1.557
(see Fig. 4)
1.542
0.224
ft= 1; f(x)=1
in the Morgenstern-Price
method.
Whitman
and Bailey give F.S. = 1.24-l .31
for Morgenstern and Price solution. It is suspected that the large
differences are due to the surfaces
not being exactly the same. See
also Bell (1968) F.S.=1*49
Fig. 3
[after
Whitman
Fig. 4
ACKNOWLEDGEMENTS
The work described in this Paper was carried out in the Civil Engineering Department of
the Imperial College of Science and Technology, London. This forms part of the main line
of research into the stability of slopes and foundations, supported by the Science Research
Council.
The calculations were carried out on the CDC 6400-6600 computer at Imperial College.
The Author gratefully acknowledges the help given by Dr N. Ambraseys and Dr D.
Papastamatiou through many discussions during the course of this work.
Delivered by ICEVirtualLibrary.com to:
IP: 86.0.218.4
On: Fri, 08 Oct 2010 12:58:16
430
S. K. SARMA
The Author is also grateful to Dr P. Vaughan who has read through the manuscript and
suggested many changes in the presentation.
The section shown in Fig. 3 and the corresponding Morgenstern-Price solution was provided by Mr Pachakis for which the Author is grateful.
APPENDIX
* (30)
all+ (13
Os = --v
2
sin (2ar-6)
(31)
If on the failure plane the amount of pore pressure developed is Au then adding Au to both sides of equations (30) and (31)
ol+Q
cr,=2+
sin (2a - #)
01+03
(I~ = --y
2
sin (2a-47
therefore
R=Z=
[1-z
sin (2a-+?I/[1
+z
sin (2.-&j]
(32)
From the Mohrs circle of stresses (Fig. 5(a)) it can also be shown that
2c cos 4
01-u3
-=sin#+--
al+%3
01+03
2 -Au sin 4
(33)
(34)
(35)
a1+o3
Using equation (33) and the invariant o1 + Q = ox + o,, and rearranging terms
g = 2 = 1 -sin
%
where
therefore
l-sinflsinfj
Ux = Oy l+sinfisin4
l+sin#sinp
Homogeneous case
Let it be assumed that within the sliding mass along the vertical axis, all planes inclined at an angle a
to the horizontal are in a state of limiting equilibrium.
In earth dams and slopes, we may approximate o, = yh and Au is defined by Au= R,yh where h is the
depth of a point from the free surface.
Substituting in equation (35)
(Jx = rh 1 -sin /3 sin $'(l - 2R.) _ 2c cos # sin /I
l+sinj?sin$
l+sinfisin&
(36)
(37)
then
E=
sin &+(G/yH)
H (Jx dh _ yHa 1 -sin p[(l-2RJ
2
l+sinfisin+
s0
cos +I
or
E = ICyHa/
STABILITY
ANALYSIS
OF EMBANKMENTS
AND
431
SLOPES
Fig. 50)
fl[( 1- 2R,) sin & -!-(4c/rH) cos f]
l+sinj3sin+
K _ 1 -sin
P, = R,yH=/2
(38)
with depth.
(39)
Then
X =
(K-RR,)F
tan
++cH]$
I.
of the slice.
......*
(K-R)Ftan#+cH
M(x)
(41)
where Xf(x) is the reciprocal of FL. The pattern of the variation of the local factor of safety at the vertical
section along the x-axis is represented by f(x). A solution to the problem reduces to the determination of
the value of X which will satisfy equation (16).
From the cases studied, it appears that f(x) = 1 gives physically acceptable solutions but the resulting value
of E computed from equation (23) may not exactly equal KHZ/2. The local factor of safety is then determined from
FL = [(E-P,)
(42)
Non-homogeneous case
To extend the method to non-homogeneous
Equation (35) may be written as
0% = ao,+b+d.Au
(424
where
I-sinflsin+
a = l+sinpsin+
b = _2ccos#sinj3
I+sinBsin#
d =
2 sin 4 sin /3
l+sinpsin+
(43)
It is assumed that, if the shear strength along the vertical section were the same as that in the failure plane,
all planes inclined at an angle a to the horizontal would be in a state of limiting equilibrium.
Then a, b
and dare constants along the vertical section and c and $ are those that apply at the slip surface.
Then
E, =
9+1
hj
0~ #a = aW(h+ ~--f#-a,(h~+
l-hj)2/2$
b(hj+l-hj)+dP,,,,
(46)
432
S. K. SARMA
Fig. 6
where
wI = weight per unit area above the level j = :z; Y~D&
DH, = h,+,-h,
Pw, = 3rd v, + v, + 1)(5+ 1- h)
height at level j, therefore
X = Z: Xj = (2 [(E,-P,,)
tan +j+cj(hj+l-hj)]}A.f(x)
(47)
(48)
by
(49)
DHj = hj+l--h,
(50)
2%
yH2/2
(51)
by
(52)
Cci. DHj
(55)
(56)
tan $+EH]
from
FL = [(E-zP,j)
tan $+EHJ/X
STABILITY
ANALYSIS
OF EMBANKMENTS
AND
SLOPES
433
Fig. 7
REFERENCES
Ambraseys, N. (1959). The seismic stability of earth dams 2, 67. PhD thesis, University of London.
Bell, J. M. (1968). General slope stability and analysis. Jnl Soil Mech. Fdns Div. Am. Sot. Civ. Engrs 94,
SM6, 1253-1270.
Bishop, A. W. (1955). The use of the slip circle in the stability analysis of slopes. Gdotechnique 5, No. 1,
7-17.
Bishop, A. W. & Morgenstern, N. R. (1960). Stability coefficients for earth slopes.
Geotechnique 10,
No. 4, 129-150.
Chugaev, R. R. (1964). Stability analysis of earth slopes. Government Production Committee for Power
USSR All Union Scientific Research Institute of Hydraulic EngineerEngineering and Electrification.
ing.
Janbu, N. (1957). Earth pressures and bearing capacity calculations by generalized procedure of slices.
Proc. 4th Znt. Conf. Soil Mech. Fdn Engng 2,207-212.
Jennings, P. (1971). Engineering features of the San Fernando Earthquake of February 9th, 1971. Earthquake Engineering Research Laboratory report, California Institute of Technology No. EERL 71-02,
pp. 302-304.
Kenney, T. C. (1956). An examination of the methods of calculating the stability of slopes. MSc thesis,
University of London.
Madej, J. (1971). On the accuracy of the simplified methods for the slope stability analysis.
Archwm.
Hydrotech. 18, No. 4, 581-595.
Morgenstern, N. R. & Price, V. E. (1965). The analysis of the stability of general slip surfaces. GPotechnique 15, No. 1, 79-93.
Nonveiller, E. (1965). The stability analysis of slopes with a slip surface of general shape. Proc. 6th Znt.
Conf. Soil Mech. Fdn Engng 2, 522-525.
Whitman, R. V. & Bailey, W. A. (1967). Use of computers for slope stability analysis. Jnl Soil Mech.
Fdns Div. Am. Sot. Civ. Engrs 93, SM4, 475-498.