Stability analysis of slopes with planar failure using variational calculus and numerical methods
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RESEARCH ARTICLE
Stability analysis of slopes with planar failure using variational calculus and numerical methods Norly BELANDRIAa* , Roberto ÚCARa, Francisco M. LEÓNb, Ferri HASSANIc a
Geological Engineering School, University of Los Andes, Mérida 5101, Venezuela Mechanical Engineering School, University of Los Andes, Mérida 5101, Venezuela c Department of Mining and Material Engineering, McGill University, Montreal, QC H3A 0E9, Canada b
*
Corresponding author. E-mail: [email protected]
© Higher Education Press 2020
ABSTRACT 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 pseudostatic load. KEYWORDS
1
slopes stability, planar failure, variational calculus, numerical methods
Introduction
Plane failure occurs due to the existence of layers or strata of different composition Although plane failure in soils is practically non-existent, it can occur in soil coverings on rocks or in rocky slopes owing to the presence of parallel discontinuities in the slope. However, the critical plane failure surface should be determined to theoretically indicate the most unfavorable plane when stabilizing a slope to rectify the instability generated from several statistically possible planes. Slope stability in soils and rock mass has been of interest to researchers of different fields. Consequently, several techniques and methods for evaluation and safety factor (SF) calculation have been developed. Limit equilibrium is one of those methods, wherein it compares the resisting forces with the mobilized forces to estimate the SF [1,2]. Article history: Received Sep 26, 2019; Accepted Jan 11, 2020
The other methods comprise numerical models that show stress distribution and displacements [3], the upper bound analysis [4–6], and the slope stability analysis considering seismic effects [7–10]. A method
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