Chen, Wai-Fah

Snitbhan, Nimitchai

English

Lehigh University: Lehigh Preserve

Lehigh University
Lehigh Preserve
Fritz Laboratory Reports Civil and Environmental Engineering
1973
On slip surface and slope stability analysis,
September 1973 (75-36)
Nimitchai Snitbhan
Wai-Fah Chen
Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-
reports
This Technical Report is brought to you for free and open access by the Civil and Environmental Engineering at Lehigh Preserve. It has been accepted
for inclusion in Fritz Laboratory Reports by an authorized administrator of Lehigh Preserve. For more information, please contact
preserve@lehigh.edu.
Recommended Citation
Snitbhan, Nimitchai and Chen, Wai-Fah, "On slip surface and slope stability analysis, September 1973 (75-36)" (1973). Fritz
Laboratory Reports. Paper 1988.
http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1988
ON SLIP SURFACE AND
SLOPE STABILITY ANALYSIS
by
Nimitchai Snitbhan
Wai F. Chen
Fritz Engineering Laboratory
Department of Civil Engineering
Lehigh University
Bethlehem, Pennsylvania
September 1973
Fritz Engineering Laboratory Report No. 355.17
ON SLIP SURFACE AND SLOPE STABILITY ANALYSIS
by
N. Snitbhan1 and W. F. Chen2
ABSTRACT
Key Words: normal stress distribution; slip surface; slope stability;
soil mechanics; variational method
The method of variational calculus is applied to obtain the
shape of slip surface and the corresponding normal stress distribution.
For a horizontal slope of uniform soil, a logarithmic spiral surface
of angle qJ is found to be the most critical surface. This contradicts
the report made earlier by Spencer. The results also emphasize the
correctness of the log-spiral failure mechanism assumed in the upper
bound method of limit analysis.
I Grad . Student, Dept. of Civ. Engrg., Lehigh Univ., Bethlehem, Pa.;
formerly Design Engineer, Fuller Company, Catasauqua, Pa.
2Associate Prof. of Civ. Engrg., Fritz Lab., Lehigh Univ., Bethlehem,
Pa.
i
"TABLE OF CONTENTS
ABSTRACT
1. INTRODUCTION
2. SOME PHYS lCAL FACTS AND THE IR SIGNIFICANCE
3. MATHEMATICAL FORMULATION OF THE PROBLEM
4. SHAPE OF SLIP SURFACE
5. NORMAL STRESS DISTRIBUTION
6. CONCLUSIONS
7 • ACKNOWLEDGEMENTS
APPENDIX I - REFERENCES
APPENDIX II
APPENDIX III - NOTATION
FIGURES
i
1
2
3
6
7
8
9
10
11
13
14
-1
1. INTRODUCTION
In the recent work of Spencer [5J on the shape of slip sur-
face in the stability analysis of embankments in uniform soil, the
following two questions were raised:
1. Is the arc of a logarithmic spiral a more critical shape
for the cross-section of the slip surface than a circular
arc?
2. If the slip surface is an arc of a logarithmic spiral in
cross-section, is its shape determined by the angle of
shearing resistance ~ of the soil?
To answer these questions, Spencer analyzed the problem using
the method of slices with the assumption of parallel inters lice forces
and concluded that (1) the circular slip surface is more critical than
the logarithmic spiral; and (2) there is no justification for assuming
that the shape of the spiral is determined by the value of the angle of
internal friction ~ of soil.
In a follow-up discussion to the paper, Chen [lJ pointed
out that the shape of slip surface and the normal stress distribution
are interrelated and both factors should be treated as the variables.
By assuming that the inters lice forces being parallel to each other in
the'method of slices, the variable on the normal stress distribution
for different shapes of slip surface was being implicitly assumed in
Spencer's analysis. Chen concluded that the proper method of analysis
to answer the above mentioned questions should therefore consider all
the shapes of the slip surfaces as well as all the possible distribution
-2
of normal stress on the slip surface. Chen, although disagreed with
Spencer's conclusion, did not present any analytical results of esta-
blishing the most critical slip surface and its associated normal stress
distribution on the surface. However, Chen did suggest that this is
an optimization problem and can be approached by applying the method
of calculus of variations. The method may require considerable mathema-
tical treatment to arrive at any appreciable solution.
The work to be described herein is directed towaras an attempt
to settle this argument. Using the method of variational calculus, the
shape and normal stress distribution of the most critical surface are
determined simultaneously.
It may be of interest to note that, by using the log-spiral
surface of angle cp, the moment equation of static equilibrium is inde-"
pendent of the normal stress distribution along the surface and, thus,
permits the stability problems to be solved in a relatively simple
mathematical form.
2. SOME PHYSICAL FACTS AND THEIR SIGNIFICANCE
A typical slope of homogenous soil under a uniform surcharge
load, q is shown in Fig. 1. The slope remains stable as long as the
stress developed within the soil mass does not exceed soil strength.
Instability is initiated as the applied load q reaches its critical
value and the collapse of the slope may be described by the rigid body
slide of soil mass along one of many "potential" surfaces, S as shown
n
in Fig. 1. At the incipient of collapse, the conditions of static
-3
equilibrium of the sliding mass
2:H = 0, 2:V = 0, rn = ° (1)
as well as the yield or failure criterion must be satisfied everywhere
along the surface. The most critical of all these potential surfaces
is theoretically the one which allows minimum applied load. In absence
of surcharge load (q = 0), the gravitational weight of the soil mass
acts solely as the external load applied on the slope.
As an example, consider a uniform slope of Fig. 2. The
positions and values of stability factors, N = H y/c for several
s c
critical slip surfaces (plane, circular and log-spiral) have been
given by Taylor [7J where H = critical height, c = cohesion and
c
y = unit weight. It is possible, then, to sketch in one figure the
three types of slip surfaces and compare the volume of the sliding
mass for each surface. This is illustrated in Fig. 3 for slopes having
base angles of ~ = 90° and ~ = 70°. The results show clearly that the
most critical shape is the log-spiral surface which also corresponds to
the minimum weight W of the sliding mass. It can, therefore, be con-
cluded that, of all the potential slip surfaces, the one which allows
the minimum weight W of the sliding mass gives the most critical situa-
tion. This condition will be used as the criterion of optimization in
the following mathematical formulation.
3. MATHEMATICAL FORMULATION OF THE PROBLEM
As stated earlier, in absence of load q the weight of the
sliding mass W is the only applied load on slope and may be defined
-4
by a functional
W= sP
w
de
s
2P = 1E- - W I(eb - e )w 2 1 0
in which WI is weight of the area O-B-A-C as shown in Fig. 2 and r(e)
is an unknown function defining the shape of the slip surface.
(2)
(3)
Referring to Eq. 1 and Fig. 2, the three equilibrium equations
can be written as
~ horizontal forces = 0 gives
S ['1" cosO' - 0' sinO'] ds = 0
s
~ vertical forces = 0 gives
S[-T sinO' - 0' cosO'] ds + W= 0
s
~ moment = 0 gives
S [0' r sins - '1" r COSs] ds + W t = 0
s
,/ r \
0' = TI - e - arctan \~)
(4)
(5)
(6)
(7a)
TI
S = "2 ( r \arctan ~) (7b)
The tangential shear stress, T and normal stress, 0' are related through
the following Coulomb failure or yield criterion
• T = C + 0' tamp (8)
Using the Coulomb criterion (8), Eqs. 4 , 5 and 6 become
eh eh eh
S PI de = 0, S P2 de = 0, S P3 de = 0 (9a,b,c)
e e e
0 0 0
-5
in which
! l
Pl = (- a) i (r cose)' tamp + (r sine)' J - c(r cose) I (10)
a r(r lP2 = cose) , - (r sine)' tamp; - c(r sine)'
+1. r 2
Wl
-
e - e2 h 0
(11)
(12)
where r(e) and aCe) are as yet two unknown functions. The problem of
finding the critical slip surface and its associated normal stress
distribution on the surface may now be stated as follows: Given the
slope shown in Fig. 2, determine the shape function r(e) and stress
function aCe) so as to minimize the weight functional, W of Eq. 2
subjected to the constraint conditions of equations (9a, b, c) . With
Lagrange's multiplier denoted by Al A2 and A3 , one can write
Since all integrands in P
w
' Pl , P2 and P3 involve only
r(e), aCe) and the first derivative of r(e), the Euler differential
equation will be first order, and can be represented by
dr-or -: or 0-, ' oa(e) =dQ j oar (e)_;
and ~r or 1 or 0de ! or' (e)J or(e) =
(13)
(14)
(15)
After substitution, integration and simplification of equations (14,15),
it follows that the two unknown functions r(e) and aCe) must satisfy
the following first-order differential equations
-6
r' [>'"2(cose - tamp sine) - Al(tancp cose + Sine)]
+ r r Al (tancp sine - cose) - A2(sine + tancp cose)J
+ (rr' - r 2 tancp) A3 = 0
independently of the normal stress distribution ~(e), and
~' [A2 (COSe - tancp sine) - Al(tancp cose + sine) + A3 rJ
(16)
+ ~(2r A3 tancp) - y r(l + A2) + A3 (2c r - y r
2 cose) = 0 (17)
The shape of the most critical slip surface can therefore be obtained
by first solving Eq. 16 for r(e). Once the fun~tion r(e) is determined
Eq. 17 can then be used for the determination of ~(e) which describes
the corresponding normal stress distribution along the critical slip
surface obtained earlier.
4. SHAPE OF SLIP SURFACE
For convenience of solution, Eq. 16 is now transformed from
polar to cartesian coordinates (Fig. 4)
A y'
- A - A3 (YY' + x)1 2
tancp IAl + A. 2Y' .,+ - A (y - xy') : = 0 (18)3 J
where x = r cose, Y = r sine
Equation (18) can also be written in the form
(Y -
A1\ ( A2,
tancp r- - (Y - Al \ ( A2\~- y' A) - x + A) + -,+ y' x+ -', = 0 (19)A/ A3 ,l..J
Let
"'2
X = x + r-,
3
-7
(20)
Equation (19) now becomes
- y'¥ - X + tan~(- Y + Y'X) = 0
By substitution into Eq. 21 the following terms
(21)
X = r cose, Y = r sine
Y' =
r cose + r' sine
r' cose - r sine
•. the complicate form of Eq. 16 now reduces to the simple form
-;2 tan~ - r r' = 0 (22)
from which r(e) = r exp(e-e )tanl:p
o 0
(23)
is the general solution. Equation (23) obviously represents the
simplest form of log-spiral surface of angle ~ having r as an abritrary
o
co.nstant.
5. NORMAL STRESS DISTRIBUTION
Rewriting Eq. 17 with respect to the new coordinates, one
obtains
cr' + 2 cr tan~ - ~ + 2c - y r cose = 0
"'3
Equation (24) is a linear, first-order differential equation from
which there exists an exact solution of the form
(24)
o(e)
Y r ) exp (8-8 )tan,!,co· 0
-- + -~---~--
tanq:J 1 + 9tan2 q:J
-8
(3 tanq:J cose + sine) + D exp(-2e tanq:J) (25)
in which the Lagrange multiplier, A3 and the integration constant, D
are as yet to be determined.
Since the moment equation (9c) is independent of cr(e), the
two remaining force equations (9a,b) may therefore be satisfied by
the proper choices of A3 and D. Substitute r(e) of Eq. 23 and cr(e)
of Eq. 25 into Eqs. 9a,b and solve for A3 and D, the final form of
the non-dimensionalized cr(e) can be expressed by
QiQl - A + r 3tanp
y H - 1 . (H/~ ) (1 + 9tan2 q:J) exp (e tanq:Jf"
o 0
r {1 exp(etanrn ) + sine exp(etanpf~ + A exp(- 2etanrn )Lcos "t' 3tanq:J -' 2 "t' (26) .
The constant terms of Al , A2 and H/~o are given in details in Appendix II.
As an illustration, Fig. 5 shows the normal stress distribution obtained
from Eq. 26, for slopes with base angles of ~ = 90° and ~ = 70° and
soil friction angle q:J = 20°.
6 • CONCLUS IONS
From the results of this work, the following conclusions
can be drawn:
1. For a horizontal slope of uniform soil, the most critical
slip surface is logarithmic spiral of angle q:J.
-9
2. The normal stress distribution is not generally independent
of the shape of slip surface. Although any assumed normal
stress distribution in the conventional methods of analysis
may give a reasonable answer, it does not always lead to the
optimum solution.
3. The rotational failure mechanism (logarithmic spiral)
utilized in the upper bound method of limit analys~ is
appropriate in the framework of limit equilibrium methods.
The results reported in Ref. [2J should therefore present
the best possible solution of the problem.
4. The variational method provides a profound and useful means
of analyzing slope stabilility problems. For the case of
complicate slope boundary and loading conditions, the mathe-
matical formulation of the problem with proper modifications
is still applicable and the numerical results can always be
obtained without much difficulties.
7. ACKNOWLEDGEMENTS
The research reported herein was supported by the National
Science Foundation under Grant GK-14274 to Lehigh University.
Many helpful suggestions made by Dr. H. Y. Fang, Director of
Geotechnical Division, and Dr. D. Edelen of the Center for the Applica-
tion of Mathematics are gratefully ~cknowledged.
The writers thank Shirley Matlock for typing the manuscript
and John Gera for the figure drafting.
-10
APPENDIX I - REFERENCES
1. Chen, W. F.
DISCUSSION ON CIRCULAR AND LOGARITHMIC SPIRAL SLIP SURFACES,
Journal of Soil Mechanics and Foundations Division, ASCE,
Vol. 96, No. 8M1, January 1970, pp. 324-326.
2. Chen, W. F., Snitbhan, N. and Fang, H. Y.
STABILITY OF SLOPES IN ANISOTROPIC NONHOMOGENEOUS SOILS,
Fritz Engineering Laboratory Report No. 355.13, Lehigh
University, Bethlehem, Pa., March 1972.
3. Hilderbrand, F. B.
METHODS OF APPLIED MATHEMATICS, Second Edition, Prentice-Hall,
Inc., Englewood Cliffs, New Jersey, pp. 119-193.
4. Kogan, B. I., and Lupashko, A. A.
STABILITY ANALYSIS OF SLOPES, Soil Mechanics and Foundation
Engineering, No.3, May-June 1970, pp. 153-157.
5. Spenc~r, E.
CIRCULAR AND LOGARITHMIC SPIRAL SLIP SURFACES, Journal of
Soil Mechanics and Foundations Division, ASCE, Vol. 95,
No. 8M1, January 1969, pp. 227-234.
6. Spencer, E.
CLOSURE ON CIRCULAR AND LOGARITHMIC SPIRAL SLIP SURFACES,
Journal of Soil Mechanics and Foundations Division, ASCE,
Vol.' 96, No. 8M4, July 1970, pp. 1466-1467.
7. Taylor, D. W.
FUNDAMENTALS OF SOIL MECHANICS, John Wiley and Sons, New
York, 1948.
APPENDIX II
The following are constant terms which must be substituted
into Eq. 26 to obtain the required normal stress distribution cr(a)
,in its non-dimensionalized form.
-11
eh
A = 1
~[- 3(1 + tan2~) u4 (2a) - 3tan~ u1(2a) - u3 (2e)]ao
1
I:h \ah " , I:hu/a) u1(a) a , . 4(H/r ) (1 + 9tan2~) U4 (a0) u1(- a)0
u3(- e)
o + 0 0
I:h "1 (-9) I::
0
\ ah lah \:h(cos~ + 17tan~) u4 (2a) eo + u1(2a) ao - 3tan~ u3(2a)
+
0
I:h4(H/r ) (1 + 9tan2~) u4 (ao) u3(- e)0
0
A =2
-12
where the functions u(e) and fare defined as
u1(e) (tancp cose + sine) exp(e tamp)
uZ(e) (tamp sine + cose) exp(e tanqJ)
u3(e) = (tanqJ sine - cose) exp(e tanqJ)
U4 (e) exp(e tanqJ)
{
exp[z(eh - eo) tanqJ] - I") r. lf = ZtanqJ(f _ f _ f) "lsmeh exp[ (eh - eo) tanqJ] - sine i1 Z 3· 0,
(3tanqJ eose + sine )1
o 0 r
sine (L/r) (Zeose - L/r )f Z = _---..;o::...-_-=-o__6__.;:;.o__~0:....
and the ratios H/r and L/r can be expressed in terms of the angles
o 0
eo and eh in the forms
and
-13
APPENDIX III - NOTATION
The following symbols are used in this paper.
c
D
I
L
q
r ,r(e),r (e),r(e)
o 0
r' (e)
s
y
"1'''2'''3
eo ,eh ,eo ,eh
O!
= constants defined in Appendix II
= cohesion
= integration constant in Eq. 25
= functions defined in Appendix II
= height and critical height of embankment, respectively
= function defined in Eq. 13
= length, see Fig. 2
= moment arms, see Fig. 2
= functions defined in Eqs. 9a,b,c
uniform surcharge load
= length variables of logarithmic spiral curve,
see Fig. 4
dr/de
slip surface
functions defined in Appendix II
weight of sliding mass, fictitious weight, see Fig. 2
= normal stress distribution along slip surface
= tangential stress distribution along slip surface
unit weight of soil
= friction angle of soil
= constant parameters
angular variables, see Fig. 4
= slope angle
angle defined in Eq. 7a and
angle defined in Eq. 7b
-14
--~
~
Fig. 1 Slope with Potential Slip Surfaces
L-15
Fig. 2 Slope of Uniform Soil
••
/
(Hdmin.
( Hclmin.
~16
-------~
.......-'VV"'V'
--- Plane
-~ --Circular
--- Log~Spiral
,------,.- -'7:~.
I /
I
I
---Plane
---- Circular
--- Log~Spiral
Fig. 3 Comparison of Weights of the Sliding Mass for Different
Slip Surfaces
y y
•
~2
~3
x
~ 8~3
X
Bh
Fig. 4 Transformation of Coordinates
-17
renlion Zone
la-II Distribution
YH
•
-o.t!
-18
•
He
-0.07
lid' Distri bution
YH
0.26
8=730
Fig. 5 Normal Stress Distribution


On slip surface and slope stability analysis, September 1973 (75-36)

https://preserve.lehigh.edu/cgi/viewcontent.cgi?article=2987&amp;context=engr-civil-environmental-fritz-lab-reports

On slip surface and slope stability analysis, September 1973 (75-36)

Abstract

Similar works

Full text

Available Versions

Lehigh University: Lehigh Preserve