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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (5) : 1262-1273
Stability analysis of slopes with planar failure using variational calculus and numerical methods
Norly BELANDRIA1(), Roberto ÚCAR1, Francisco M. LEÓN2, Ferri HASSANI3
1. Geological Engineering School, University of Los Andes, Mérida 5101, Venezuela
2. Mechanical Engineering School, University of Los Andes, Mérida 5101, Venezuela
3. Department of Mining and Material Engineering, McGill University, Montreal, QC H3A 0E9, Canada
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This study investigates the technique of variational calculus applied to estimate the slope stability considering the mechanism of planar failure. The critical plane failure surface should be determined because it theoretically indicates the most unfavorable plane to be considered when stabilizing a slope to rectify the instability generated by several statistically possible planes. This generates integrals that can be solved by numerical methods, such as the Newton Cotes and the finite differences methods. Additionally, a system of nonlinear equations is obtained and solved. The surface of the critical planar failure is determined by applying the condition of transversality in mobile boundaries, for which various examples are provided. The number of slices is varied in one of the examples, while the surface of the critical planar failure is determined in the others. Results are compared using analytical methods through axis rotations. All the results obtained by considering normal stress, safety factors, and critical planar failure are nearly the same; however, in this research, a study is carried out for “n” number of slices using programming methods. Sub-routines are important because they can be applied in slopes with different geometry, surcharge, interstitial pressure, and pseudo-static load.

Keywords slopes stability      planar failure      variational calculus      numerical methods     
Corresponding Author(s): Norly BELANDRIA   
Just Accepted Date: 24 July 2020   Online First Date: 20 October 2020    Issue Date: 16 November 2020
 Cite this article:   
Norly BELANDRIA,Roberto ÚCAR,Francisco M. LEÓN, et al. Stability analysis of slopes with planar failure using variational calculus and numerical methods[J]. Front. Struct. Civ. Eng., 2020, 14(5): 1262-1273.
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Roberto ÚCAR
Francisco M. LEÓN
Fig.1  Failure mechanism passing through the slope toe.
Fig.2  Representation of the contour conditions in the crest of the slope, considering the mechanism of planar failure.
parameters description
n, r number of slices in which the sliding mass is subdivided
d parameter used for the first two slices
hb integration constant to solve the integrals numerically of the sum of horizontal forces, vertical forces, and moments
Dx width of the slice
Ψ coefficient of friction (tanf)
c cohesion
sn normal stress in each slice investigated
un interstitial pressure for each slice
kh, kv horizontal and vertical seismic coefficients
pu unit weight
yn ordinate of the potential surface of failure in each slice investigated
gn ordinate of the points belonging to the surface of the slope for each slice
yp = g′(x) derivative of the surface of the slope with respect to the axis of the abscissas
Tab.1  Parameters used in the programming
Fig.3  Representation of the investigated slope, indicating the geometrical characteristics and the stresses obtained for each slice. (a) Slope: 4 slices; (b) Slope: 8 slices.
item equations system analytical slides
Matlab ESS Maple rotation axis (Fellenius slide)
s0 36.9231
s1 32.3382 32.31 32.3078 32.3077 32.3077
s2 23.0464 23.08 23.0767 23.0769 23.0769
s3 13.8157 13.85 13.8459 13.8461 13.8461
s4 4.6457 4.615 4.6155 4.6153 4.61538
s5 7.492e−15
SF 0.9998 0.9997 0.9997 0.9997 0.9983
xc 342.929 1.919e6 59682.043
yc −883.640? −4.957e6 −1.5415e5
Tab.2  Comparison of the results from some programs and the analytical method
item Maple slices= 4 Maple slices= 6 Maple slices= 8 Maple slices= 10
s1 32.3077 34.0977 34.6163 35.0787
s2 23.0765 27.6405 30.0001 31.3852
s3 13.8457 21.3380 25.3841 27.6920
s4 4.6154 15.1865 20.7685 23.9991
s5 9.1828 16.1531 20.3065
s6 3.3233 11.5380 16.6141
s7 6.9232 12.9221
s8 2.3087 9.2304
s9 5.5390
s10 1.8479
xc 53642.015 61.516 18104.948 11283.951
yc −138557.275 −156.795? −4673.631 −29144.725
Tab.3  Maple programming results for different numbers of slices
stress (kPa) rotation axis slices= 4 rotation axis slices= 6 rotation axis slices= 8 rotation axis slices= 10
s0 36.9231 36.9231 36.9231 36.9231
s1 32.3077 33.8462 34.6154 35.0770
s2 23.0770 27.6924 30.0001 31.3847
s3 13.8462 21.5385 25.3847 27.6924
s4 4.6154 15.3846 20.7693 24.0000
s5 7.492e−15 9.2308 16.1539 20.3077
s6 3.0769 11.5385 16.6154
s7 7.492e−15 6.9231 12.9231
s8 2.3077 9.2308
s9 7.492e−15 5.5385
s10 1.8462
s11 7.492e−15
Tab.4  Results of the analytical method for different numbers of slices
Fig.4  Representation of the slope studied.
item values non-dimmentional values
s1 490.18 s1/gh 0.68
s2 371.41 s2/gh 0.52
s3 252.61 s3/gh 0.35
s4 133.78 s4/gh 0.18
xc −20276.13? xc/h −675.81
yc 275575.41 yc/h 9185.85?
SF 1.156 SF 1.156
Tab.5  Values of normal stresses acting on the potential failure curve, SF, and geometric center
Fig.5  Iterations of the planar failure. (a) Variation of the failure surface and the SF; (b) variation of the SF and the failure angle α (SF min and α critical); (c) coordinate of the slope’s crest.
normal stresses (kPa) geometric center, coordinate of the crest and width of the slice (m) SF
s1 s2 s3 s4 xc yc x5 Dx
1.36 3.95 6.55 9.14 46.75 54.35 40.5 10.12 1.95
Tab.6  Results from applying the plane surface failure mechanism
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