Three-Dimensional Stability Analysis of ConvexSlopes in Plan View
Orang Farzaneh1; Faradjollah Askari2; and Navid Ganjian3
Abstract: A method of three-dimensional slope stability analysis for convex slopes in plan view is presented here based on theupper-bound theorem of the limit analysis approach. The method can also be used to determine the bearing capacity of foundationsadjacent to such slopes. A rigid-block translational collapse mechanism is considered in which energy dissipation takes place along planarvelocity discontinuities. Comparing the bearing capacity of foundations, numerical results indicate that the one located near convex slopeshas less capacity than the one located near straight slopes. Inversely, slopes not subjected to surcharge loads are more stable when theyare convex. Concerning the bearing capacity analyses, consideration of the curvature effect of convex slopes is more significant infrictional soils whereas for slope stability analyses, mentioned effect is of prime importance in cohesive soils. Numerical results ofproposed algorithm are presented in the form of nondimensional graphs.
DOI: 10.1061/�ASCE�1090-0241�2008�134:8�1192�
CE Database subject headings: Slope stability; Limit analysis; Algorithms.
Introduction
The assessment of slope stability has received much attentionacross geotechnical communities because of its practical impor-tance. Numerous methods have been proposed for slope stabilityanalysis. In general, these methods can be classified into the fol-lowing types: �1� limit equilibrium approach which is the mostcommon; �2� numerical solutions based on continuum mechanics;and �3� limit analysis approach.
Stability problems of slopes are often analyzed by methodsbased on two-dimensional models, neglecting the end effects ofthe failure mechanism. However, the failure regions of actualslopes usually have finite dimensions and therefore a three-dimensional �3D� approach is more appropriate to analyze suchstability problems. 3D slope stability problems fall into threecategories:1. Slopes that are subjected to loads of limited extent at the top.2. Slopes in which the potential failure surface is constrained by
physical boundaries, such as a dam in a narrow rock-walledvalley.
3. Slopes with nonplanar surfaces such as road embankments at
1Assistant Professor, Civil Engineering Dept., Univ. College ofEngineering, Univ. of Tehran, Tehran, Iran �corresponding author�.E-mail: [email protected]
2Assistant Professor, Geotechnical Eng. Dept., International Instituteof Earthquake Engineering and Seismology. E-mail: [email protected]
3Ph.D. Student of Geotechnical Engineering, Civil Engineering Dept.,Univ. College of Engineering, Univ. of Tehran, Tehran, Iran. E-mail:[email protected]
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curves, or mining waste where the granular material heapshave well-defined corners.
Most analyses for slope stability have dealt with straightslopes with a planar surface. However, there are many convexslopes in plan view with nonplanar surfaces. During the past de-cades, the influence of plan curvature on the stability of slopeshas been investigated mainly by Giger and Krizek �1975, 1976�,Leshchinsky and Baker �1986�, Baker and Leshchinsky �1987�,Xing �1988�, and Ohlmacher �2007� for some special cases. Gigerand Krizek �1975, 1976� used the upper-bound theorem of limitanalysis to study the stability of a vertical corner cut subjected toa local load. They assumed a kinematically admissible collapsemechanism and, through a formal energy formulation, assessedthe stability with respect to shear strength of soil. Leshchinsky etal. �1985� presented a 3D analysis of slope stability based on thevariational limiting equilibrium approach and proved that it canbe considered as a rigorous upper bound in limit analysis. Lesh-chinsky and Baker �1986� used a modified solution of the ap-proach mentioned to study 3D end effects on stability ofhomogeneous slopes constrained in the third direction and appliedit to investigate the stability of vertical corner cuts. Using a varia-tional approach, Baker and Leshchinsky �1987� discussed the sta-bility of conical heaps formed by homogeneous soils. Xing�1988� proposed a 3D stability analysis for concave slopes in planview using the equilibrium concept. Based on the limit equilib-rium method, Ohlmacher �2007� investigated a case study includ-ing convex and concave slopes.
Michalowski �1989� introduced a rigorous 3D approach in thestrict framework of limit analysis for homogeneous and straightslopes. In his analysis, the geometry of slope and slip surface wasunrestricted and both cohesive and frictional soils were included.Farzaneh and Askari �2003� improved Michalowski’s algorithm inthe case of 3D homogeneous slopes and extended it to analyze thestability of nonhomogeneous slopes. This paper is concerned witha solution of the 3D stability problem of convex slopes in planview �Ganjian 2003� which is a development of the method pro-
posed in primary works �Farzaneh and Askari 2003�.
The present study is based on the kinematic approach of limitanalysis and can be used to find both the limit load and the factorof safety of a convex slope subjected to its self-weight and aconcentrated surcharge load.
The limit analysis method is based on the extensions of themaximum work principle derived by Hill �1948�, and was givenin the form of theorems by Drucker et al. �1952�. Using thisapproach, the true solution from a lower bound to an upper boundwould be bracketed. Applicability of this theorem requires thatthe soil’s behavior be perfectly plastic and the deformation begoverned by the normality rule. Considering the theorem of kine-matic approach �upper bound�, the rate of work done by tractionand body forces would be less than or equal to the energy dissi-pation rate in any assumed kinematically admissible failuremechanism.
Here, the geometry of the convex slopes is modeled by a sec-tor of conical surface and that of the corners with two planarsurfaces. Fig. 1 shows the plan and cross section of the collapsemechanism �in the plane of symmetry� for a curved slope.The collapse mechanisms consist of rigid translational motionblocks which are separated by planar velocity discontinuity sur-faces and surrounded by one lateral surface on each side. Fourblocks are considered in the mechanism shown while the numberof blocks would be increased if necessary. As can be seen, theexternal load is assumed to be uniformly distributed over a rect-angular area.
The basic developments applied to primary algorithms, pro-posed by Michalowski �1989� and Farzaneh and Askari �2003�,
Fig. 1. Mechanism used for analysis of convex slopes �circular inplan� in current study
concern the following domains: �1� construction of failure mecha-
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nism; �2� calculation of the required parameters such as area andvolume of blocks; and �3� optimization procedure.1. Construction of failure mechanism: if the failure mechanism
was to be modeled by one of the primary algorithms, twodistinct type of blocks could be considered:a. Type-I: the blocks which include nonplanar exterior;
andb. Type-II: the blocks which are far from the surface of
slope and are similar to those in planar surface slopes.Using primary algorithms, failure mechanisms usually in-
clude several Type-I blocks with various configurations.Definition of the geometry and calculation of the areas andvolumes of Type-I blocks would be very complex and timeconsuming. The main issue of the current algorithm is theintroduction of a frontal block �block No. 0 in Figs. 1 and 2�,which limits the number of Type-I blocks to one, associatedwith a new optimization parameter. The assumption of asingle front block spanning the full width of the mechanismincludes some limitations where the convex front face com-prises a corner with a small angle.
In a particular case, the assumed mechanism can be ex-tended laterally so that none of the blocks has lateral �end�surfaces. In such cases, 3D effects are limited to the convexshape of the slope and end effects are omitted. The resultsrelated to theses cases are also represented in this paper.
2. Calculation of areas and volumes of blocks: using the mecha-nism described, the analytical geometry relations for deter-mination of the lateral planes of more blocks, as well as thesurface areas and the volumes of block Nos. 1–N-1 are simi-lar to the primary algorithm. The volume and surface areas ofthe frontal block �block No. 0� can be determined numeri-cally knowing the equations of the surface GDFGDE �Part ofa cone�, plane DHG �lateral plane�, and plane HGGH �baseplane�.
3. Optimization procedure: the present numerical technique tofind the failure mechanism corresponding to the least upper-bound load or the least upper estimation of the safety factoris somehow different from the procedure used in earlierworks. In this algorithm, in addition to angles �k, �k, and �,the parameter y0 is defined as a variable determining thewidth of failure mechanism that should be optimized.
The formulation of the current approach is the same as that ofearlier works. According to the associative flow rule, velocityincrement vectors across velocity discontinuity surfaces are in-clined to those surfaces at the internal friction angle �. In Fig. 1,angles �k, �k, and � �angle of lateral surfaces of block No. 0related to horizontal level �not shown�� are assumed to be knownand the failure mechanism is easily constructed.
Having found the velocities as well as the volumes and thesurface areas of all blocks, one can write the energy balance equa-tion. For the mechanism considered, energy dissipation is limitedto velocity discontinuities including the base and side faces ofeach block and the planes between adjacent blocks. The proce-dure was explained precisely by Michalowski �1995�.
Comparison with Other Results
The results of the current approach can be compared with those ofother investigators for straight slope �linear in plan view� casestending the plan curvature radius of slope to infinity. Differentmethods have been proposed for 3D analysis of straight slopes by
Baligh and Azzouz �1975�, Hovland �1977�, Chen and Chameau
�1983�, Ugai �1985�, and Leshchinsky and Baker �1986�. Com-paring the current results with most of these, good agreement isfound among them.
Ugai �1985� extended Baker and Garber’s 2D variational lim-iting equilibrium approach to 3D cohesive slopes. Leshchinskyand Baker �1986� extended a modified solution of variational ap-proach in 3D stability of slopes which has been proved by them tobe equivalent to the upper bound solution in the framework oflimit analysis.
Fig. 3 shows the ratio F3D /F2D �FiD is the safety factor in iDanalysis� as a function of d /H �d is the maximum width of theoptimum failure mechanism� obtained by Ugai �1985�, Leshchin-sky and Baker �1986�, Farzaneh and Askari �2003� and thepresent solution. As it is seen, the results of current solution andthose of Farzaneh and Askari �2003� are approximately the sameand they are in good agreements with the Ugai and Leshchinskyresults. Comparing the results with Leshchinsky’s indicates that
Fig. 2. Failure mechanism based on current a
Fig. 3. Comparison of present solution with those of Ugai,Leshchinsky, and TRASS for vertical slope in cohesive soil
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for vertical slopes with d /H greater than 5, the results are lesssensitive to the assumed shape of the end surface defining thefailure.
Giger and Krizek �1976� presented an upper-bound solutionfor the stability of a vertical cut with a variable corner angle.Leshchinsky and Baker �1986� compared the results of their pro-posed variational approach for stability of vertical corner cutswith those of Giger and Krizek �1976� as the stems from limitanalysis. Fig. 4 shows a comparison between those results and thecurrent approach. For corners with �=90°, the results are closeand current approach gives slightly better results �smaller factorof safety �Fs��. For straight vertical slopes ��=180° �, Giger andKrizek’s results are more conservative, while Leshchinsky andBaker’s results are the best, and the current approach gives resultsslightly close to those.
m for analysis of convex slopes in plan view
Fig. 4. Comparison of stability factors obtained using currentapproach with those of Giger and Krizek, Leshchinsky and Baker �forvertical cut�
Baker and Leshchinsky �1987� presented some numerical re-sults for 3D stability of conical heaps without the top part usingthe variational approach. Fig. 5 shows comparisons of thoseresults as the only available results for rounded slopes in theliterature with the current approach. The relationship betweenFs�3D�
/Fs�2D�and R /H for various combinations of material
strength �i.e., various �; �=c /� .H . tg �� are presented. It canbe seen that the 3D safety factor decreases with the growth ofR /H, in both approaches, although the current results are slightlygreater.
Numerical Results
The mechanisms and formulations presented in this paper can beused to calculate the stability number of a convex slope or acorner cut under their own weights as well as to determine thebearing capacity of a rectangular shallow foundation located ad-jacent to such slopes.
Stability of Convex Slopes
A symmetrical model �Fig. 6� including five rigid blocks is usedto calculate the stability number of homogeneous convex slopesin plan view. Changing the relative curvature radius of slope�R0 /H�, a series of graphs was obtained through this 3D method.Figs. 7–10 illustrate the ratio of stability numbers obtained from3D analysis to stability numbers obtained from 2D upper-boundanalysis �indicated in Table 1�. The dimensionless parameters pre-sented are defined as
Fig. 5. Comparison of present solution with those of Baker andLeshchinsky for rounded slope with b=60
Fig. 6. Failure mechanism of convex slopes used in analyses
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Fig. 7. Stability charts for convex slopes with �=30°: �a� ��c=0; �b���c=1; �c� ��c=3; and �d� ��c=10
where Fs�factor of safety obtained by analysis. The parametersd, H0, and R0 are shown in Fig. 6.
These analyses are performed by assuming that there are re-strictions in sides which prevent extending collapse mechanismsand constructing mechanism without “end planes” as shown inFig. 6. If there is no restriction �stability of conical heaps�, amechanism without “end planes” will result in less stability fac-tors. Table 2 presents the stability numbers of conical heaps with-out end planes �called extended� compared with those ofrestricted convex slopes with d=2*R0.
The influence of plan curvature and width of the mechanismon the stability of convex slopes can be derived from these nu-merical results as follows:1. Convex slopes in plan view are more stable than straight
slopes. In general, the smaller the ratio R0 /H0 is, the higherthe stability of convex slope in plan view.
2. Generally, decreasing ��c, three-dimensional effects aremore significant. In other words, the effect of curvature ofslope is more important in cohesive soils.
3. The effect of curvature on the stability of convex slopes isless for steeper slopes.
Stability of Corners „Slopes with Two-Planar Surfaces…
Using the current approach, the stability number of two-facedslopes with various corner angles is obtained. The failure mecha-nism made by five blocks with parameters d, H, and � is shownin Fig. 11. The results are presented in Figs. 12–14 as the ratioof stability number obtained from 3D analysis to 2D analysis�Table 1�. From the general pattern of these curves, it is apparentthat Ns decreases with the increasing corner angle ���. Resultssimilar to those described above �for slopes which are sectors ofconical surfaces� can be observed in Figs. 12–14 for corners.
Bearing Capacity of Foundations near Convex Slopes
The effect of curvature on the bearing capacity of shallow rect-angular foundations on slopes is studied. The failure mechanismused in these analyses consists of four rigid blocks �Fig. 15�.
Figs. 16 and 17 exhibit the ratio of the bearing capacity ob-tained from the current 3D algorithm to that of a strip foundationusing the Farzaneh–Askari approach as presented in Table 3. ��c
is defined by Eq. �2� and other parameters are shown in Fig. 15.The influences of plan curvature on bearing capacity of foun-
Table 1. 2D Stability Numbers �Ns�2D��
�
��c 30 45 60 90
0 9.02 7.28 6.28 4.01
1 12.81 9.84 7.98 4.36
2 16.15 — 9.45 6.14
3 18.77 16.76 10.65 6.34
10 40.70 28.88 21.65 12.13
Fig. 10. Stability charts for convex slopes with �=90°: �a� ��c=0;�b� ��c=1; �c� ��c=3; and �d� ��c=10
1. The bearing capacity of shallow foundations on convexslopes is less than that of shallow foundations on straightslopes. In other words, the smaller the ratio R /L is, thesmaller the bearing capacity of foundations located on con-vex slopes.
2. Increasing ��c, the influence of curvature on the bearing ca-pacity of foundations on slopes increases and the effect ofcurvature is of prime importance in frictional soils.
3. The effect of curvature on the bearing capacity is more sig-nificant for steeper slopes.
Table 2. �Ns�3D / �Ns�2D of Conical Heaps without End Planes �Extended
0
Beta�deg� R0 /H Extended Restricted
30 1 1.258 1.492
2 1.164 1.328
3 1.133 1.274
5 1.100 1.177
60 1 1.097 1.418
2 1.065 1.321
3 1.056 1.197
5 1.052 1.150
90 1 1.173 1.391
2 1.193 1.292
3 1.200 1.238
5 1.209 1.131
Fig. 11. Failure mechanism of corners used in stability analyses
Fig. 12. Stability charts for corners with �=30°
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Examples
The usefulness of the stability charts presented could be bestdemonstrated through two examples: one for corners and one forconical heaps.
pared with Those of Restricted Convex Slopes with d=2*R0
Consider a vertical corner cut with angle �=120° and height of12 m �Fig. 11�. The properties of the soil are: �=17.0 kN /m3;�u=0, and cu=45 kPa. The width of the corner is restricted to30 m �d=30 m�. The factor of safety of the potentially slidingmass is needed.
Using Fig. 14 for �=90° and ��c=0, one gets �Ns�3D /�Ns�2D=1.42 for d /2H=1 and �Ns�3D / �Ns�2D=1.25 for d /2H=2.Interpolating the results, for d /2H=1.25, �Ns�3D / �Ns�2D will be
Fig. 15. Failure mechanism used in bearing capacity analysis offoundation on convex slope
Fig. 16. Bearing capacity charts for convex slopes with �=60°: �a���c=0; �b� ��c=1
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equal to 1.29. From Table 1, �Ns�2D based on the upper-boundanalysis is 4.01 and therefore �Ns�3D=1.29*4.01=5.17. Knowing�Ns�3D and considering Eq. �1�, the safety factor of the slope willbe equal to 1.14.
Critical Height of Curved Slope in Plan View
A 40 m width U shaped valley will be filled with mining waste.The frontal slope of the fill can be considered as a sector ofconical heaps with slope �=60° and R0=50 m �as Fig. 6�.The properties of the fill are: �=18.5 kN /m3; �=10°; andc=32.5 kPa. What is the critical height of slope?
The critical height �corresponding to Fs=1� will be calculatedas follows:
A first value should be considered for H0. AssumingH0=10 m, to use Fig. 9, one computes ��c=18.5*10 /
Table 3. Bearing Capacity Ratio of Strip Foundation near Slopes
q2D /c �=60 �=90
a /B ��C=0 ��C=1 ��C=0 ��C=1
0.2 3.14 4.90 — —
0.5 3.50 5.82 2.42 1.7
1.0 4.02 7.20 2.84 2.1
2.0 — — 3.31 4.3
Fig. 17. Bearing capacity charts for convex slopes with �=90°: �a���c=0; �b� ��c=1
32.5*tan 10° =1.0 and gets �Ns�3D / �Ns�2D=1.14 for d /2H=2and R0 /H=5.0. From Table 1, the 2D stability number �Ns2D
�based on the upper-bound analyses is 7.98. The result is that�Ns�3D=1.14*7.98=9.10. Knowing �Ns�3D and consideringEq. �1�, the 3D safety factor of the convex slope will be 1.60.
The calculation will be iterated with other values of H0 toobtain the critical height of the slope. In this example, the itera-tion results in the critical height of about 20 m.
Conclusions
A three-dimensional slope stability algorithm for the locallyloaded or unloaded convex slopes in plan view is described, basedon the upper-bound theorem of limit analysis. The analysis allowsassessment of either the safety factor or the bearing capacity offoundations located on corners or convex slopes which are sectorsof conical surfaces. The method provides an upper bound solutionfor both cohesive and frictional soils. The collapse mechanism isgenerated in the form of rigid blocks by extending algorithmsbelonged to Michalowski �1989� and Farzaneh and Askari �2003�.
When a convex slope is analyzed three dimensionally usingthe rigid block method, the shape of some blocks may be affectedin various manners by the geometry of slope. A definition ofgeometry of such blocks and calculation of their volumes andsurface areas requires complicated and time consuming proce-dures. The key issue in the present development is introducing afrontal block, which is the sole block of failure mechanism beingaffected by the convexity of the slope, associated with a newoptimization parameter. This concept results in a considerable re-duction of the time required for calculations.
Comparing the current results with the available upper-boundsolutions for vertical corner cuts or cone heaps, it shows goodcompatibility. The results are presented in the form of nondimen-sional graphs. The main conclusions based on these results can bederived as follows:1. Comparing the bearing capacity of foundations, numerical
results indicate that the one located near convex slopes hasless capacity than the one located near straight slopes. In-versely, not subjected to external surcharge load, convexslopes would be more stable.
2. Concerning the bearing capacity analyses, consideration ofthe curvature effect of convex slopes is more significant infrictional soils, whereas Regarding slope stability analyses,consideration of the effect mentioned in cohesive soils is ofprime importance.
Notation
The following symbols are used in this paper:a � distance of slope crest from beginning of
loading;B � loading width;c � soil cohesion;
d � maximum width of failure mechanism;
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H � height of slope;k � block number;L � loading width;N � total number of blocks;
Ns � stability factor;qu � average limit pressure;R0 � radius of conical surface in top of slope;� � corner angle of slope;
�k, �k, �k � geometrical parameters of failure mechanism;� � angle of slope;� � specific weight of soil;
�c� � defined as ���h /c�tan ��; and� � internal friction angle.
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