Effect of The Axis of Moment Equilibrium in Slope Stability Analysis
Effect of The Axis of Moment Equilibrium in Slope Stability Analysis
Effect of The Axis of Moment Equilibrium in Slope Stability Analysis
-
D. G. FREDLUND
Department of Civil Engineering, University of Saskatchewan, Saskatoon, Sask., Canada S7N OW0
Z . M. ZHANG
Beijing Geotechnical Institute, Beijing, China
AND
L. LAM
Clifton Associates, Calgary, Alta., Canada
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Some of the methods of slices satisfying moment equilibrium derived for circular slip surfaces have been extended
to accommodate noncircular (or composite) type slip surfaces. A question arises regarding the point about which moment
equilibrium should be taken and whether varying the center for moment equilibrium has a significant effect upon the
computed factor of safety. This paper addresses the question of the effect of the center for moment equilibrium as
it pertains to noncircular (or composite) slip surfaces. In particular, extensions of the Ordinary, Bishop's simplified,
and the General Limit Equilibrium (GLE) methods are examined. The results show that considerable variations in the
factor of safety can occur when using the extended Ordinary method. The extended Bishop's simplified method shows
varying factors of safety as the moment axis moves vertically. Variations in the computed factor of safety can generally
be expected to be less than 12%. The GLE, Morgerstern-Price, and Spencer methods are independent of the axis for
moment equilibrium.
Key words: slope stability, limit equilibrium, moment equilibrium, factor of safety, noncircular slip surface.
Quelques-unes des mCthodes des tranches satisfaisant l'tquilibre des moments derive pour les surfaces de glissement
circulaires, ont CtC Clargies pour accommoder les surfaces de glissement de type non-circulaire ou composite. Une ques-
For personal use only.
tion se pose quant au point par rapport auquel 1'Cquilibre des moments devrait Ctre calculC et quant B savoir si le fait
de faire varier le centre d'kquilibre des moments a un effet significatif sur le coefficient de sCcuritC. Cet article pose
la question concernant l'influence de la position du centre de l'equilibre des moments dans le cas des surfaces de glisse-
ment non-circulaires ou composites. L'on examine en particulier les mCthodes ordinaire et simplifite de Bishop, et
celle de l'equilibre limite gCneral (GLE). Les risultats montrent que des variations considerables du coefficient de sCcuritC
peuvent se produire lorsque la mCthode ordinaire Clargie est utilisee. La mkthode simplifiee elargie de Bishop montre
des coefficients de skurite qui varient lorsque l'axe des moments bouge verticalement. L'on peut s'attendre a ce que
des variations dans le coefficient de sCcuritC calcule soient gCnCralement infkrieures a 12%. Les mkthodes GLE, de
Morgenstern-Price et de Spencer sont independantes de la position de l'axe de 17Cquilibredes moments.
Mots elks : stabilitC de talus, Cquilibre limite, Cquilibre des moments, coefficient de sCcuritC, surface de glissernent
non-circulaire.
[Traduit par la redaction]
the moments for all slices can be summed. From this expres- I(a)
sion, the factor of safety with respect to moment equilibrium + Center ot rzatlon
(76, 55)
'
can be derived. For the simplified methods of slices with
circular slip surfaces, it has become acceptable to use the
center of rotation as the axis of moments. The reasonableness
of this choice has unfortunately never been investigated.
The objectives of this paper are (i) to study the effect of
the position of the moment axis on the factor of safety (three
methods of slices, namely the Ordinary, Bishop's simplified,
and the GLE methods, have been selected to show the effect
of the axis of moments); and (ii) to conduct a quantative
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x-coordinate
investigation to ascertain the most reasonable position for
the axis of moments when using Bishop's simplified method
for composite slip surfaces. + Center of rotation
(76, 55)
General evaluation
The effect of the axis of moments on the factor of safety
is studied for three methods, namely the Ordinary (Fellenuis
1936), Bishop's simplified (Bishop 1955)' and GLE (Fredlund
et al. 1981) methods. All these allow the use of a common
axis for the summation of moments. 30 I I I I I I
50 60 70 80 90 100 110
A single homogeneous slope as shown in Fig. la is used
as the reference case. A case with a composite slip surface x-coord~nate
(i.e., slip surface consisting of circular arcs and a straight FIG. 1. Geometry of example problems.
I
line) illustrated in Fig. l b is used for a comparative study.
The study was conducted using the PC-SLOPE software devel-
For personal use only.
l oped by GEO-SLOPE International Ltd. (Fredlund 1985). The percentage of difference is not a minimum when the
I By using the GLE formulation, this package makes it possi- axis of moments is assumed to be the center of rotation.
I ble to obtain the factors of safety for all three methods.
Each case was analyzed by systematically changing the
Rather, the region of smallest difference is about 5-6 m to
the right of the center of rotation. This is true for slopes
position of the axis about which moments were taken. The with a composite slip surface and slopes with a completely
water table in all cases was assumed to be below the slip circular slip surface.
surface. More than 900 stability analyses were performed (2) For Bishop's simplified method, the factor of safety
for various centers of moments, and the factors of safety is affected where the axis for moment equilibrium is changed
for the various methods were determined. The computed in the vertical direction (Figs. 5 and 6). This finding has been
factor of safety at each moment center was compared with verified by all the cases analyzed. Using Bishop's simplified
the factor of safety obtained by the GLE method. The fac- method for circular slip surfaces shows that the rotation
tor of safety computed using the GLE method should be center and the moment center gave the same factor of safety
essentially unaffected by the axis about which moments are as that of the GLE method. As shown in Fig. 6, the best
taken, since all elements of statical equilibrium are satisfied. moment center was not coincidental with the rotation center
As shown in Fig. 2, the percentage difference was essentially for the circular portion of a composite slip surface. The
O%, except near the edge where there are numerically results indicate that the Bishop's simplified method can be
unstable zones. The empty space in Fig. 2 represents the used for most cases, provided a reasonable axis of moments
numerically unstable zones. When the axis of moments was is selected.
chosen in this region, the net resistant moment due to the The above findings led to a further study on the effect
unstabilized shear force was so small that the factor of safety of the location of the axis of moment equilibrium.
equation becomes unstable and cannot achieve convergence.
The effect of the axis of moments for the Ordinary and the
Bishop's simplified methods were determined in terms of Effect of the location of the axis on moment
the percentage difference in the factor of safety between each equilibrium for Bishop's simplified method
method and the GLE method at each of the moment centers. Bishop's simplified method is probably the most widely
The detailed contours of the percentage difference in the used of all the simplified methods. This is not only because
factors of safety are presented in Figs. 3-6. The results verify of its easy application to relevant engineering problems but
that the percentage difference for all axis positions is essen- also, more importantly as pointed out above, because it gives
tially zero for the GLE method (Fig. 2). The results from a factor of safety similar to that of the rigorous methods,
Bishop's simplified and the Ordinary methods are contoured such as Morgenstern-Price, Spencer's, or GLE methods.
(Figs. 3-6). The following conclusions can be drawn from This is true for all cases where the slip surface is circular.
these figures. This method was extended to the case of a composite slip
(1) For the ordinary method, the factor of safety changes surface by Fredlund and Krahn (1977). A derivation of the
widely with changing the location of the axis of moments. factor of safety equation for Bishop's simplified method for
The factor of safety is significantly affected when the axis a composite slip surface is presented in the Appendix. How-
of moments changes in a horizontal direction (Figs. 3 and 4). ever, a question arises regarding the uniqueness of the solu-
CAN. GEOTECH. J. VOL. 29, 1992
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For personal use only.
x-coordinate
FIG. 2. Percentage difference of the factor of safety at different axes of moments for the reference case using the GLE method.
Numbers in the graph indicate percent difference between various axes.
FIG. 4. Contours of the percentage difference in the Ordinary FIG. 6. Contours of the percentage difference of Bishop's
method factor of safety at different axes of moment equilibrium simplified method factor of safety at dieferent axes of moment equi-
for the non circular case referenced to the GLE method. librium referenced to the GLE method.
For personal use only.
FIG. 8. Effect of the radius R and the parameter d for the case of horizontal bedrock.
For personal use only.
must be less than 1.5. Correspondingly, the effects of R and When the c l / y H ratio increases to a larger value (i.e., up
d/R on dy/R for c l / y H of 0.025, 0.050, and 0.075 are to 0.075), a larger radius of the slip surface will likely be
shown in Fig. 8. used. The radius can become an important factor affecting
Smaller c' / y H values yield shallower critical slip surfaces the dy value. For example, the following observations can
(i.e., lower D ' / H values), and therefore low dy/R values be made from Fig. 8: (i) when R = 20 m, D 1 / H = 1.5 and
can be expected. For all cases shown in Fig. 8 where the dy,,, = 2.32 m at d / R = 0.25; (ii) when R = 25 m,
D ' /H ratio is less than 1.5 (or R < 20 in the example), dy/R D 1 / H = 2.0 and dy,,, = 9.85 m at d/R = 0.32; and
is always less than 0.15. Where D ' / H = 1.25, even smaller (iii) R = 30 m, D ' / H = 2.5 and dy,,, = 19.76 m at
dy values were observed. If the rotation center is selected d/R = 0.44.
as the moment center, the difference in the factor of safety However, even for these cases, the larger dy values only
when compared with the GLE method will be less than about appear for highly composite conditions. When the slip sur-
2-3% (refer to Fig. 16). This is obviously negligible. face is slightly composite, for example, when d/R < 0.15,
FREDLUND ET AL.
18 a l p =OOO
+ d p =023
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0 00 020 0 40 060
d/R
FIG. 10. Effect of bedrock tilt for D 1 / H = 2.0.
to Fig. 16).
Effect of bedrock tilting
A 1:2 slope with c' = 15 kPa and 6 ' = 20" was selected
to study the effect of the bedrock tilting. Some 55 cases,
including a variety of intersections d and tilting angles a ,
were analyzed for conditions of D r / H = 1.5 and 20.
The results in Figs. 9-1 1 show that the best position of
the axis for moment equilibrium was strongly influenced by
the tilting of the bedrock. Variation in the results also
occurred for different D r / H and d/R values.
Quantitatively, the major factor that influences dy/R is
the ratio of the bedrock tilting angle a to the slope angle 0
(Fig. 11). The dy distance increases as the bedrock tilting
angle CY increases. When CY/Pis equal to about 0.4, dy is at
its maximum value. When CY continues to increase, dy
decreases. When a = ,B (i.e., the bedrock is parallel to the FIG. 11. Effect of bedrock tilt for various d / R values.
slope surface), dy = 0. In other words, for all semi-infinite
slopes, Bishop's simplified method can be applied by select-
ing the rotation center as the moment center. Data shown in Figs. 12 and 13 and Table 1 indicate that,
Also, for all slightly composite slip surfaces, even when although flatter slopes tend to have a larger dy value to get
the bedrock is not horizontal, there will not be a significant the same factor of safety as that of the GLE method, the
error if the rotation center is selected as the moment center. change of dy value is not particularly sensitive to the slope
For example, from Fig. 11, when d/R < 0.2, for any CY steepness when compared with the influence of other factors
value, dy/R is less than 0.4 and the error in this factor of such as R, a , and d.
safety is less than about 8%. The data shown in Figs. 12 and 13 are for the cases with
D r / H = 1.5 and 2.0. If the rotation center is used as the
Effect of slope steepness moment center, most percentage differences in factor of
A series of geometrics with slopes of 1.5: 1, 2: 1, 3: 1, and safety will be less than 2% and no larger than 5% for
4: 1 was studied for the purpose of evaluating the effect of
extreme cases.
the steepness of the slope on the moment equilibrium axis.
The bedrock tilt angle a was assumed to be zero for all cases. Effect of soil parameters
The slope length, H/(tan P), which is a projection of the Different combinations of shear strength parameters (i.e.,
slope surface in the horizontal direction, appears to be a c' and 6' ) were used to show the influence of shear strength
good means of normalizing the effect of slope steepness on the axis of moment equilibrium. The results indicate that
(Figs. 12 and 13). the effective cohesion parameter has a stronger influence
CAN. GEOTECH. J. VOL. 29, 1992
0
t
2:1 SLOPE, R = 20, H -
3:l SLOPE, R = 20, H = 10
10
0 00 0 04 0 08 0 12 0 16 0 20 0 24 0.26
dlR
FIG. 12. Effect of slope steepness for D ' / H = 1.5.
For personal use only.
dlR
FIG. 13. Effect of slope steepness for D 1 / H = 2.0.
on dy than does the friction angle 4' (Fig. 14). The dy value Discussion of the results
increases with an increase in cohesion (Fig. 15). In total, 133 slopes geometries ranging from 1.5: 1 to 4: 1
For the example shown in Fig. 15, when cohesion changed were studied. A wide range of geometries and soil param-
from 0 to 15 kPa, dy increased only from 1.2 to 2.4 m. This eters was selected for various aspects of the study. These
is of no significance in terms of the error in the factor of variables varied as follows: R = 17.5, 20, 25, and 30 m;
safety. The same conclusion can also be drawn by comparing S = 5-10 m; cr = 0-26.7'; /3 = 18-34"; c ' = 0-15 kPa;
the curves shown in Fig. 8. The curves were close together 4 ' = 10-20"; and c l / y H = 0.025, 0.050, and 0.075.
when D 1 / H was equal, even though the values of c l / y H An attempt was made to assimilate all the data in terms
were quite different. of dy/R versus the percentage difference in the factor of
FREDLUND ET AL.
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0 00 0 04 0 08 0 12 0 16 0 20 0 24 0 28
d/R
FIG. 14. Effects of c' and 4' values on variable dy for a 2:l slope with D f / H = 1.5 and R = 20.
For personal use only.
c' (kPa)
FIG. 15. Effect of c' value on the variable dy.
safety (i.e., the difference in the factor of safety if the rota- centage difference in the factor of safety was never larger
tion center for the circular portion is used as the moment than 8%.
center). The results show the overall influence of all factors
mentioned (Fig. 16). The percentage difference in the factor
of safety increases with an increase in dy. The percentage Conclusions
approaches a limit at larger dy values. For all the 133 slope (1) Methods of slices satisfying complete statical equilib-
geometries analyzed, the percentage difference in the factor rium (e.g., the Morgenstern-Price, Spencer's and GLE
of safety was no larger than about 12%; the highest dy value methods) are not affected by the position of axis of
exceeded 20 m. The higher percentage difference in the fac- moments. On the contrary, all simplified methods satisfying
tor of safety appeared when the radius was large. When R I moment equilibrium are affected by the position of the axis
20 m; D 1 / H 5 1.5; with the other factor varying, the per- of moments. This study shows that for the Ordinary method,
CAN. GEOTECH. J. VOL. 29, 1992
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FIG. 16. Percent difference in the factor of safety using the rotation center as center for moment equilibrium.
rotation.
values
(4) It is possible to extend the utilization of Bishop's
simplified method to slopes with a composite slip surface.
dy (m) at the following slopes
The results showed that the center of rotation of the circular
d (m) 1.5:l 2:l 3:l 4:l portion can be used as the center for moment equilibrium,
with the extreme error in the factor of safety being less than
12%. Except in the cases with D' / H = 1.25, negligible dy
values were observed. All the factors of safety obtained by
using the center of rotation are on the conservative side.
(5) For a shallow slip surface (for example, R < 20 m
NOTE:The dy were obtained from the slope or D ' / H I 1.5), the error was negligible (less than 2-3%)
with the variables S = 5-10 m, H = 5-10 m, for cases with horizontal bedrock. The differences were
R = 20 m, c' = 15 kPa, and q5' = 20". never larger than 8% for any combination of the geometry
of slope and the slip surface. For deep-seated, slightly com-
posite slip surfaces (R I 30 m, d/R I 0.4), the error in
the best position of the axis for moment equilibrium is not the factor of safety was less than 8%.
the same as the center of rotation for the slip surface. This
is also true for the simplest slope geometry with a circular
slip surface. Bishop, A.W. 1955. The use of the slip circle in the stability anal-
(2) When Bishop's simplified method is used, the same ysis of slopes. Gkotechnique, 5: 7-17.
Bishop, A.W., and Morgenstern, N.R. 1960. Stability coefficients
factor of safety can be obtained as with the GLE method for earth slopes. Geotechnique, 10: 129-150.
by using the center of rotation as the axis moment equilib- Duncan, J.M., and Wright, S.G. 1980. The accuracy of equilib-
rium, when (i) the slip surfaces are circular; and (ii) the rium methods of slope stability analysis. Proceedings of the Inter-
geometry is that of a semi-infinite slope (e.g., bedrock national Symposium on Landslides, Delhi, vol. 4, pp. 247-254.
parallels the slope surface). Fellenius, W. 1936. Calculation of the stability of earth dams. Pro-
(3) For other conditions, the best axis for moment equi- ceedings, 2nd Congress on Large Dams, vol. 4, pp. 445-466.
librium is different than the center of rotation for the cir- Fredlund, D.G. 1985. PC-SLOPE slope stability analysis user's
cular position. For all the cases analyzed using Bishop's manual, S-30. CEO-SLOPE International, Ltd., Calgary, Alta.
simplified method, the results verified that the factor of Fredlund, D.G., and Krahn, J. 1977. Comparison of slope stability
safety was only affected by moving the axis of moments in methods of analysis. Canadian Geotechnical Journal, 14:
429-436.
a vertical direction. Therefore, dy alone can be used to define Fredlund, D.G., Krahn, J., and Pufahl, D.E. 1981. The relation-
the position of the moment axis corresponding to the dif- ship between limit equilibrium slope stability methods. Proceed-
ference between the center of rotation for the circular por- ings, 10th International Conference on Soil Mechanics and Foun-
tion and the center of moment equilibrium. The analyses dation Engineering, Stockholm, vol. 3, pp. 409-416.
also showed that for all cases of a composite slip surface, Morgenstern, N.R., and Price, V.E. 1965. The analysis of the
positive dy values were obtained. In other words, the best stability of general slip surfaces. Gkotechnique, 15: 70-93.
FREDLUND E T AL.