Two-dimensional Limit Equilibrium Research Articles

Application of bursting pressure and conventional drained behavioural theory to the Brumadinho failure

The Brumadinho tailings dam failure in 2019 has had a profound impact on the way tailings engineers analyse and interpret stability, with the focus of assessments moving away from conventional drained behaviour to consideration of undrained and post-liquefaction resistance. One of the contributing reasons is that the Expert Panel investigation of the failure concluded that, despite the drained factor of safety being adequate, the failure took place in an undrained manner. This paper examines the Brumadinho failure from a conventional drained perspective using two-dimensional limit equilibrium and finite difference modelling. Collectively, these analyses show that the failure can also be explained by a drained triggering mechanism and therefore suggest that several additional lessons may be learnt from the failure.

A Case Study for Stability Analysis of a Toppling Bank Slope with Fault Fracture Zones Developed under the Action of Bridge Loads and Reservoir Water

The mountainous areas of Southwest China have the characteristics of valley deep-cutting, a large topographic gradient, complex geological structures, etc. With the development of infrastructure construction in the area, the construction of bridges across valleys has gradually increased, and the phenomenon of slope failure occurs more and more frequently. As the weak interlayer, the fault fracture zones have a significant influence on the geological structure and stability of slopes, while the complexity of the mechanism of the deformation and failure of slopes increases with the combination of the development of the fracture zones and toppling deformation. This paper took the toppling bank slope of bridge foundations developed with fault fracture zones in Lancang River as the research object. Through an on-site field survey and geological survey technologies, it identified the distribution range of the fracture zones on the bank slope and determined the characteristics of the rock mass in the fracture zones. A stability evaluation model for the bank slope of the bridge foundations was established using the limit equilibrium method and discrete element method. Based on the two-dimensional limit equilibrium analysis, the potential failure modes of the bank slope were explored, and the stability of the bank slope under bridge loads was calculated. Through the three-dimensional geological model of the bank slope, including the fracture zones and toppling bodies, the three-dimensional discrete element numerical simulation method was adopted to simulate and calculate the deformation and failure process of the bank slope under different bridge loads and working conditions. According to the calculation results, the influence of bridge loads and reservoir water on the stability of the bank slope was analyzed from the perspectives of displacement, plastic zone, stability coefficient, and other factors. The formation process of the plastic zone and the development of the sliding surface were revealed, the incentive mechanism of bridge loads and reservoir water on the deformation and failure of the bank slope was analyzed, and the influence of fault fracture zones on the stability of the bank slope and the development of toppling deformation was determined. The results indicate that the fault fracture zones are important geological structures that affect the deformation and failure of the bank slope as a weak interlayer. Under the influence of bridge loads and reservoir water, the stability of the bank slope is affected by the quality of the rock mass and the development of the fault fracture zones, resulting in the unmet need for safety requirements and maybe leading to instability. Based on the calculation results of the stability evaluation prediction model for the bridge foundation bank slope and the engineering geological conditions, the bridge scheme has been selected.

How does mine tailings slurry solids concentration affect stability of dam embankment slope?

ABSTRACT In view of many past failures of tailings storage facilities, practising engineers have been analysing current design guidelines for different elements of these facilities, including for tailings dams. In this paper, an attempt is made to investigate how varying tailings slurry solids concentration ( c s ) affects the stability of tailings dam embankment slope. To achieve this, elaborate two-dimensional limit equilibrium and finite element simulations were conducted and a steady-state seepage analysis was performed so as to obtain accurate free surface water flux through the slope. Initially, numerical analyses were performed for the downstream embankment slope when no material is retained upstream. Subsequently, with all parameters kept constant, the same slope was then checked when water is retained upstream, and finally for varying slurry solids concentration. Special attention was paid to the boundary between sedimentation and consolidation, defined as the solids concentration limit beyond which effective stresses start to develop. From the stability analyses performed, it is found that the factor of safety of embankment slope varies nonlinearly with total unit weight of the slurry. For coal tailings slurry, factor of safety reduced from 1.24 when water is retained to 1.15 when tailings slurry with c w = 60% is retained, representing a drop by 7.3%.

Three-dimensional strength reduction and two-dimensional limit equilibrium methods of slope stability analysis

Landslides caused tremendous damage to the life and properties of society. In this study, cut slopes were analyzed using the two-dimensional (2D) limit equilibrium method (LEM) by slope/w software and a three-dimensional (3D) strength reduction method (SRM) by using Fast Lagrangian Analysis of Continua in 3 Dimensions (FLAC3D), which is a finite difference method (FDM) numerical software. Analysis of the slope is done by representative soil samples collected and transported to the laboratory to determine input parameters like unit weight (γ), shear strength parameters (c,φ), and the water level position. Strength parameters were determined from direct shear tests for cohesionless soil and consolidated undrained (CU) triaxial tests for cohesive soils. For 2D analysis, the factor of safety (FOS) was calculated for the condition of dry and maximum water levels using Morgenstern-price methods, and the analysis gave FOS 0.743 and 0.614 for the dry and maximum water level conditions, respectively; for SRM, the input parameters of bulk and shear modulus (B, G) which were determined from the empirical equation using modulus of elasticity(E) and Poisson's ratio(ʋ).Moreover, ʋ were adjusted according to the principle of φ-v inequality. For 2D analysis, FOS were 0.743 and 0.614 for dry and maximum water levels using Morgenstern-price methods, whereas 3D analysis gave FOS 0.34 and 0.31 for dry and maximum water levels, respectively. From the analysis, it is concluded that 2D LEM is given higher FOS than 3D FDM. Moreover, the 3D analysis showed that velocity and displacement in the lateral Y direction were minimal and maximum in the horizontal X direction. The maximum horizontal velocity for the respective dry and maximum water level conditions gave 6.64*10−4 m/step and 6.04*10−4 m/step and maximum horizontal displacement 7.36 m and 6.06 m for dry and maximum water levels, respectively.

Stability prediction and optimal angle of high slope in open-pit mine based on two-dimension limit equilibrium method and three-dimension numerical simulation

The optimization design of the open pit slope can ensure safety in production at the same time bring considerable economic benefits, so it is necessary to evaluate the stability of the initial excavation scheme of the high slope in an open-pit mine and optimize the initial slope angle according to the evaluation results. In this paper, an exhaustive study of Anyi open-pit mine in mouding County Yunnan province China has been made, and the following results have been achieved: (1)Confirmed the geological factors which affected the slope stability of open-pit mine; (2)Quantified the stability of the open-pit mine slope; (3) Under the premise of safety, the angle of the open pit high slope has been optimized on the base of the calculations of the open-pit mine high slope stability so as to ensure the maximum economic benefit. In this paper, the study area was divided into six zones. Firstly, qualitative analysis of slope stability was made by analogy method of engineering geology and stereographic projection method according to the orebody location, the properties of rock and soil in slope, the height of slope, and so on. The analysis shows that the preliminary design of slope angle is so large to ensure safety. Secondly, the stability of different slopes of a high slope is the quantitative analysis of different zones in high slope was made by two-dimensional limit equilibrium method and the finite difference method. In this paper, the mechanical parameters of rock are obtained by a laboratory test, and the stability of the high slope is quantitatively analyzed by two-dimensional limit equilibrium and three-dimensional numerical simulation.

Accuracy of Two-Dimensional Limit Equilibrium Methods in Predicting Stability of Homogenous Road-Cut Slopes

Although limit equilibrium methods are widely used by engineers and scientists in predicting the stability of homogenous slopes, their use has been demonstrated to present significant errors due to the violation of kinematic and static admissibility. The concern is often voiced regarding the accuracy of limit equilibrium methods (LEMs) solutions in predicting the stability of homogenous slopes. There are no exact limit equilibrium solutions or charts available that could be used to check the LEMs solutions. The present study has used the rigorous upper and lower bounds solutions of limit analysis based on finite element formulations of the bound theorems to benchmark and develop an accuracy classification chart of limit equilibrium methods in predicting the stability of the homogenous slope. Six case studies of homogenous road-cut slopes that vary with material properties were used and the effect of the increase in material strength with depth was considered. The results of LEMs and limit analysis solutions have shown that Janbu simplified limit equilibrium solutions are closely related to those of rigorous upper bound solutions with an accuracy error ranging from 1 to 7% in various slope materials. Meanwhile, the Corp of Engineer 2 limit equilibrium solutions were found to overestimate among other methods, with an accuracy error ranging from 12 to 17% in various cases. Based on the results of the study an accuracy error classification chart of LEMs is developed.

Methodology for Resloping of Rock Slope Using 3D Models from UAV-CRP Technology

Assessing the stability of rock slopes is essential to ensuring the typical performance of adjacent transportation infrastructures but simulating the existing field topography and boundary conditions for stability analysis is challenging. This paper describes a case study in which highly weathered rock slopes adjacent to a railroad track in Texas were analyzed using close-range photogrammetry data obtained from an unmanned aerial vehicle with close-range photogrammetry (UAV-CRP). A highly weathered rock mass with low intact strength was considered as a single continuum that undergoes circular failure and was analyzed using Spencer’s two-dimensional (2D) and three-dimensional (3D) limit equilibrium method (LEM) analyses. A 3D critical slip surface was obtained to identify the critical sections on which to conduct the individual 2D stability analysis and obtain a safe resloping angle. These angles were further used to reslope the whole rock slopes and conduct a 3D analysis to arrive at a safe and economic slope angle for each side of the rock-cut. The realistic geometrical slopes developed from the aerial imagery were used to develop a framework for comprehensively analyzing the stability of a rock slope undergoing circular failure.

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

The rock slope stability analysis can be performed using deterministic and probabilistic approaches. The deterministic analysis based on the safety concept factor uses fixed representative values for each input parameter involved without considering the variability and uncertainty of the rock mass properties. Probabilistic analysis with the calculation of probability of failure instead of the factor of safety against failure is emerging in practice. Such analyses offer a more rational approach to quantify risk by incorporating uncertainty in the input variables and evaluating the probability of the failure of a system. In rock slope engineering, uncertainty and variability involve a large scatter of geo-structural data and varied geomechanical test results. There has been extensive reliability analysis of rock slope stability in the literature, and different methods of reliability are being employed for assessment of the probability of failure and the reliability of a slope. Probabilistic approaches include Monte Carlo simulation (MCS), the point estimate method (PEM), the response surface method (RSM), first- and second-order reliability methods (FORMs and SORMs), and the first-order second-moment method (FOSM). Although these methods may be complicated, they provide a more complete definition of risk. Probabilistic slope stability analysis is an option in most commercial software; however, the use of this method is not common in practice. This paper provides an overview of the literature on some of the main probabilistic reliability-based methods available for the design of the rock slope in open pit mining. To demonstrate its applicability, the paper investigates the stability of a rock slope in an open pit mine in the Goldfields region, Western Australia. Two different approaches were adopted: deterministic stability analysis using two-dimensional limit equilibrium and finite element shear strength reduction methods using SLIDE and RS2 software, respectively, and probabilistic analysis by applying the MCS and RSM methods in the limit equilibrium method. In this example, the slope stability analysis was performed using the Spencer method with Cuckoo search optimization to locate the critical slip surface. The results obtained were compared and commented on.

Ultimate pullout capacity of single vertical plate anchors in sand

This paper presents a limit equilibrium solution for calculating the ultimate pullout resistance of shallow vertical plate anchors in sand, considering the combined effects of anchor embedment ratio, sand shear strength, sand-anchor interface strength and anchor aspect ratio. The geometry of the failure mechanism in the solution reflects observations from model tests. The ultimate failure load of a strip plate anchor was then determined by global equilibrium analysis of the passive and active failure soil wedge. The two-dimensional limit equilibrium solution is extended for rectangular plate anchors by introducing the shape factors of Ovesen and Strømann. The solution is compared with a database of 102 pullout model tests of vertical plate anchors in sand assembled from the literature and some limit analysis results. The comparison results suggest that the proposed solution can provide a reliable analytical tool for the ultimate pullout capacity prediction of shallow vertical plate anchors buried in sand.

An Integrated InSAR-Borehole Inclinometer-Numerical Modeling Approach to the Assessment of a Slow-Moving Landslide

ABSTRACT We use results of satellite-based interferometric synthetic aperture radar, Global Positioning System, and borehole inclinometer data to constrain numerical models that improve understanding of slope deformation at the Alexandria landslide, British Columbia, Canada. Surface monitoring data and borehole slope inclinometer measurements provide important insight into the slope failure mechanism. We initially analyzed the data in a geographic information system (GIS) to create thematic maps of the landslide area (hillshade, slope, and aspect), to identify key geological features, and to produce an engineering geomorphology map of the landslide. The monitoring data and the geological/engineering geomorphological models provide important constraints for two-dimensional landslide limit equilibrium and finite difference analyses. The initial limit equilibrium analysis improved understanding of the sliding surfaces. The finite difference models were then used to simulate and investigate the potential slope deformation mechanism. The combined slope monitoring/modeling results show that the Alexandria landslide is a slow-moving, multiple-block, retrogressive slope failure. The close agreement between the limit equilibrium and finite difference analyses, together with the satellite and ground-based slope monitoring and GIS data, highlight the importance of using a multidisciplinary/integrated approach in landslide studies.

Stability Analysis of Mineral Structures using FLAC Software

The safety factor for slopes (FS) is traditionally determined using two-dimensional limit equilibrium (LEM) methods, however, the safety factor of a slope can also be calculated by FLAC software with the technique of reducing soil shear strength in the time stages until the slope fails. In this presentation, we first describe the numerical methods of stability analysis, finite difference method and FLAC software, and then we analyze the static stability using FLAC software.

A two-dimensional limit equilibrium computer code for analysis of complex toppling slope failures

Evaluation of blocky or layered rock slopes against toppling failures has remained of great concern for engineers in various rock mechanics projects. Several step-by-step analytical solutions have been developed for analyzing these types of slope failures. However, manual application of these analytical solutions for real case studies can be time-consuming, complicated, and in certain cases even impossible. This study will first examine existing methods for toppling failure analyses that are reviewed, modified and generalized to consider the effects of a wide range of external and dead loads on slope stability. Next, based on the generalized presented formulae, a Windows form computer code is programmed using Visual C# for analysis of common types of toppling failures. Input parameters, including slope geometry, joint sets parameters, rock and soil properties, ground water level, dynamic loads, support anchor loads as well as magnitudes and forms of external forces, are first loaded into the code. The input data are then saved and used to graphically draw the slope model. This is followed by automatic identification of the toppling failure mode and a deterministic analysis of the slope stability against this failure mode. The results are presented using a graphical approach. The developed code allows probabilistic introduction of the input parameters via probability distribution functions (PDFs) and thus a probabilistic analysis of the toppling failure modes using Monte-Carlo simulation technique. This allows calculation of the probability of slope failure. Finally, several published case studies and typical examples are analyzed with the developed code. The outcomes are compared with those of the main references to assess the performance and robustness of the developed computer code. The comparisons demonstrate good agreement between the results.

Three-dimensional seismic stability of slopes reinforced by soil nails

Seismic stability of slopes reinforced with soil nails has been traditionally investigated by two-dimensional limit equilibrium method. In this paper, the strength reduction method in combination with the kinematic approach of limit analysis is employed to assess the factor of safety (Fos) of slopes reinforced by soil nails using the three-dimensional rotational failure mechanism. The pseudo static approach is used to represent the seismic effects. Both the tensile failure and pull-out failure of soil nails are considered in the computations of internal energy dissipations. Comparisons are made to validate the proposed approach, which shows that the implemented approach is an efficient design tool for evaluating the factor of safety of slopes reinforced by soil nails. Parametric analysis is conducted to discuss the influence of model parameters, including nail length, nail density, soil shear strength and seismic forces, on slope stability. A set of stability charts is finally provided for fast assessments of slope safety.

Study on Stability of Deformation Body of Material Yard Slope of a Pumped Storage Power Station

The stability of granite deformation body in a pumped storage power station is studied in the paper, and the research includes field geological survey data collection, analysis of distribution law of structural plane, study on failure model of deformed body and prediction of deformation body stability. Based on that, the failure modes and stability of deformed bodies are studied in detail. According to the regional topographic map of deformed body, a two-dimensional limit equilibrium model is established. The three-dimensional model is established by using 3DEC discrete element software. The stress, strain and displacement under natural and rainstorm conditions are calculated and analyzed respectively, and the accuracy of the calculated results is determined by comparing with the monitoring data, so as to predict the deformation direction and rate of the deformed body. The longest movement distance of the deformed body along the shear outlet of different elevations is simulated separately to determine the affected area, which provides a reliable basis for making preventive measures for the construction site.

Investigation and modeling of direct toppling using a three-dimensional distinct element approach with incorporation of point cloud geometry

Block toppling instability can be a common problem in natural rock masses, especially in mining environments where excavation activity may trigger discontinuity-controlled instability by modifying the natural slope geometry. Traditional investigations of block toppling failure consider classic kinematic analyses and simplified two-dimensional limit equilibrium methods. This approach is still the most commonly adopted, but the simple two-dimensional conceptual model may often oversimplify the instability mechanisms, ignoring potential critical factors specifically related to the three-dimensional geometry. This paper uses a three-dimensional distinct element method approach applied to an example case study, identifying the critical parameters that influence direct toppling instability in an open pit environment. Terrestrial laser scanning was used to obtain detailed three-dimensional geometrical information of the slope face geometry for subsequent stability analyses. A series of sensitivity analyses on critical parameters such as friction angle, discontinuity shear and normal stiffness, discontinuity spacing, and orientation was performed, using simple conceptual three-dimensional numerical modeling. Results of the analyses revealed the importance of undertaking three-dimensional analyses for direct toppling investigations that allow identification of critical parameters. A three-dimensional distinct element analysis was then performed using a more realistic complex volumetric mesh model of the case study slope which confirmed the previous modeling results but also identified unstable blocks in high slope angle areas, providing useful information for life of mine design. The paper highlights the importance of slope geometry and fracture network orientation on potential slope instability mechanisms.

Stability Evaluation of a Toppling Deformed Body in Miaowei Hydropower Station

By analyzing the instability characteristics of deformed body after impoundment and using limit equilibrium algorithm and UDEC discrete element simulation, the failure mechanism and stability of deformed body are studied in the paper. According to the deformation degree and instability characteristics of the deformed body in different storage periods, the mechanism of instability is analyzed. Based on the regional topographic map, a two-dimensional limit equilibrium model is established to calculate the potential failure surface range and slope stability factor of QD18 deformed body under the conditions of 1314m, 1364m and 1401m water level storage. And the displacement nephogram, velocity nephogram and rock block deformation map of the deformed body under the condition of 1401 m water level are simulated by using the discrete element software, and the stress changes of each part of the deformed body after water storage are analyzed, and the failure mechanism is summarized.

Investigation of landslide failure mechanisms adjacent to lignite mining operations in North Bohemia (Czech Republic) through a limit equilibrium/finite element modelling approach

Understanding the impact of data uncertainty is a fundamental part of ensuring safe design of manmade excavations. Although good levels of knowledge are achievable from field investigations and experience, a natural geological environment is subject to intrinsic variability that may compromise the correct prediction of the system response to the perturbations caused by mining, with direct consequences for the stability and safety of the operations.Different types of geoscientific evidence, including geological, geomorphic, geotechnical, geomatics, and geophysical data have been used to develop and perform two-dimensional Limit Equilibrium and Finite Element Method stability analyses of a lignite open-pit mine in North Bohemia (Czech Republic) affected by recent landslides. A deterministic-probabilistic approach was adopted to investigate the effect of uncertainty of the input parameters on model response. The key factors affecting the system response were identified by specific Limit Equilibrium sensitivity analyses and studied in further detail by Finite Element probabilistic analyses and the results were compared. The work highlights that complementary use of both approaches can be recommended for routine checks of model response and interpretation of the associated results. Such an approach allows a reduction of system uncertainty and provides an improved understanding of the landslides under study. Importantly, two separate failure mechanisms have been identified from the analyses performed and verified through comparisons with inclinometer data and field observations. The results confirm that the water table level and material input parameters have the greatest influence on the stability of the slope.

Stability analysis of a recurring soil slope failure along NH-5, Himachal Himalaya, India

The repetitive soil slope failure along the National Highway (NH)-5 in Jhakri region of Himachal Pradesh, India draws frequent concern due to heavy damage and traffic disruption almost every year. Being only linking route from border district to the nearby land area, stability of the road-cut slopes along this highway is of major concern in regard to safe transportation. Absence of any previous stability investigation of this recurring slope failure calls for an integrated geotechnical and numerical approach in order to understand the instability factors. The geotechnical analysis has been performed to determine the inherent properties of soil materials which affect the stability of existing slope. An event-specific antecedent rainfall threshold has been suggested to quantify the relationship between rainfall and slope failure. A two-dimensional limit equilibrium method has also been executed to visualize the scenario of pre- and post-failure stability of the slope. On the basis of limit equilibrium analysis, it has been inferred that slope geometry is a major affecting parameter that influences the failure pattern. Moreover, preventive measures through benching and soil nailing have also been proposed and validated through limit equilibrium analysis for long-term stability and safe transportation.

GIS and limit equilibrium in the assessment of regional slope stability and mapping of landslide susceptibility

Slope stability analysis is an enduring research topic in the engineering and academic sectors. Probability-based analysis is the most popular method for evaluating a wide range of natural slope instability risks, but the lack of landslide data limits its application. A new method of regional slope stability analysis is presented that combines spatial analysis and two-dimensional slope analysis. First, a region is divided into slope units based on terrain and hydrology analysis, and the main potential slip sections of slope units are determined. Second, the potential slip surface of each section is determined and stability factors calculated with a two-dimensional slope analysis model. Finally, regional slope stability analysis is determined from the results of slope unit segmentation and slope stability analysis. A loess gully zone in Jintai District, Baoji City, China, constitutes the study area. A topographic map is transformed into a digital elevation model and evaluation zones determined by slope unit segmentation. A 3D geological model is devised from survey, drilling and investigation data. Geological profiles are generated automatically and slope stability analysis performed immediately with the small tool developed herein. Lastly, a map of regional slope assessment is produced. The method proposed combines the two-dimensional limit equilibrium method and spatial analysis, which are the methods usually used in engineering practice. It can provide not only a distribution map of safety factors, but also the scale and location of most dangerous slip surfaces. This study provides a new way to assess the natural slope risk of large areas.

Statistics for the calculated safety factors of undrained failure slopes

Due to the uncertainty and variability involved in ground conditions and analytical methods, the calculated factor of safety (FS) of a slope may not be exact. To know whether the calculated FS is unbiased and also the magnitude of its variability, this paper re-analyzes 43 real cases of undrained slope failure reported in the literature. The FS is re-calculated using two-dimensional (2-D) limit equilibrium methods (LEM) (i.e., the simplified Bishop's method and the Spencer's method) to minimize the human factor in FS calculation. Since all cases failed, the conditions (such as geometries) right before failures were adopted to simulate the nearly failure condition in the LEM. Based on the statistical results of the calculated FS obtained from hypothesis testing, it can be concluded that (a) the 2-D LEM seems to give unbiased FS estimates; (b) the FS variability for natural slopes is significantly larger than that for man-made slopes; (c) the conversion of the undrained shear strength (su) to its field value is the most crucial, and the modeling of su spatial variability is also crucial; and (d) the standard deviation of the human error in the logarithm of the calculated FS is about 0.25, and the human error appears to be uncorrelated to the logarithm of the calculated FS. Finally, the relationship between the failure probability of a man-made undrained clay slope and its calculated FS is developed to facilitate reliability analysis and reliability-based design.