Limit Equilibrium Model 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.

GIS-based statistical and limit equilibrium models in the assessment of slope stability and landslide susceptibility: the case study of the Aomar Miocene basin, Bouira, Algeria

GIS-based statistical and limit equilibrium models in the assessment of slope stability and landslide susceptibility: the case study of the Aomar Miocene basin, Bouira, Algeria

Study of Slope Stability of the Mining Wall in an Open-Pit Coal Mine by the Paste Cut-and-Backfill Method

According to this study’s findings, slope stability problems in open-pit coal mines can be avoided, and mine wall collapse can be effectively mitigated by the use of cut-and-backfill mining techniques. The main research results are as follows: (1) The stope and waste rock’s geotechnical, physical, and mechanical characteristics were gathered and examined; the geotechnical and mechanical characteristics found in this study largely satisfy the criteria for slope stability analysis. (2) Cemented paste backfill (CPB) materials were made of mine waste rock and fly ash at a desired ratio, mixed with cement as a bond material, and were tested in the laboratory, using a combination of cement percentages of 6%, 8%, and 10% for the cement content and 25%, 30%, 35%, and 40% for the fly ash content, to determine the ideal mix for artificial ground support in underground mines, taking into account both economic and performance factors. (3) By using this model, the changes in CPB strength were investigated under various factors influencing the cement ratio, and limit equilibrium modeling was used with the FLAC-Slope 8.1 program with different cement paste backfill ratio to calculate the factor of safety for each cement percentage after 1 day, 3 days, 7 days, and 28 days of curing time (CT) to obtain the optimum compressive strength and shear straight of cemented paste backfill with high paste fill shear strength on the slope. (4) The research results are of great significance for the safety of important facilities in open-pit mines and provide a basis for the design and safety implementation of open-pit slope engineering.

Limit equilibrium model for active failure of large-size rectangular tunnel face considering effect of height–width ratio

The stability of a rectangular tunnel face (RTF) has received less attention than that of circular tunnels; in particular, the stability of RTFs with different height–width (H/W) ratios has not yet been investigated comprehensively. In this study, extensive numerical simulations are conducted to investigate the limit support pressure (Pa) of the active failure of an RTF with different H/W ratios and to determine the appropriate failure mechanism without assuming the failure shape in advance. The result shows that, as the H/W ratio increases, Pa decreases gradually for specified heights and overburden depths of the RTF. The failure shape of the soil in front of the RTF is determined by analyzing the contours at plastic points of the soil. Subsequently, three types of failure modes, namely, partial semicircular, semicircular, and supersized semicircular modes, are identified based on the progression of plastic points in the soil. Corresponding failure models are established using the limit equilibrium method. Compared with relevant theoretical models, the proposed model predicted results that are more consistent with the numerical simulation results. Furthermore, a parametric study is performed to investigate the stability of the RTF. Finally, the predicted results are compared and analyzed with the pressure monitored inside the chamber, thus providing a theoretical reference for the jacking construction of the Shasan project.

Limit equilibrium models for passive failure of a large-diameter shield tunnel face in reinforced soft clay

Limit equilibrium models for passive failure of a large-diameter shield tunnel face in reinforced soft clay

The impact of assumptions on deterministic and probabilistic analyses of earth fill embankments

This paper considers the impact of assumptions made when formulating analytic approaches to deterministic and probabilistic analyses of slope stability. The assumptions reviewed were presented in a paper, Waddell (2023) that presented semi-quantitative stability assessment using survey data, deterministic and probabilistic slope stability analyses of fill embankments constructed from residual soil and weathered bedrock in Western Sydney. A single variable fill strength was adopted, that was obtained from an Infinite Slope analysis assuming fully saturated fill. Rainfall records were used to predict risk of saturation to calculate annual probability of slope failure. The impacts of assumptions used to calculate the strength parameter, factor of safety and predicted failure probability are addressed in this paper. Saturation assumptions, the choice of limit equilibrium model, shape of the probability distribution functions, and spatial correlation of material properties are considered.

Slope stability analysis using kinematic, limit equilibrium, and finite element models at Dessie town, northern Ethiopia

Slope stability analysis using kinematic, limit equilibrium, and finite element models at Dessie town, northern Ethiopia

Limit Equilibrium Models for Tunnel Face Stability in Composite Soft-Hard Strata

The tunnel face stability in composite strata, commonly referred to as the soft upper and hard lower condition, is a critical challenge in tunnel construction. The soft–hard ratio (SA) strongly influences the limit support pressure as well as the failure mechanism experienced by a tunnel face. This study focused on the Xingye Tunnel project in the Xiangzhou District of Zhuhai City. By conducting numerical simulations, the impact of different SAs on the limit support pressure was investigated. Furthermore, a limit equilibrium model was established on the basis of the analysis of the results of numerical simulation. The findings were then compared and analyzed alongside those of relevant theoretical models. In the event of tunnel face instability of composite strata, the deformation tends to be concentrated mainly in the soft soil layer, with less noticeable deformation in the hard rock layer. The investigation of different SAs revealed a linear decrease in the limit support pressure ratio of the tunnel face in composite strata as SA decreases. The self-stability of the tunnel face was observed when SA ≤ 0.125. Moreover, the limit support pressure ratio predicted by the truncated log-spiral model (TLSM) exhibited a higher degree of agreement with the results of numerical simulation than those of other relevant models. The superiority of TLSM was mainly demonstrated in the range of SA = 0.25 to 1.0.

Calibration of the limit equilibrium pillar failure model using physical models

The limit equilibrium model, used in displacement discontinuity codes, is a popular method to simulate pillar failure. This paper investigates the use of physical modelling to calibrate this model. For the experiments, an artificial pillar material was prepared and cubes were poured using the standard 100 mm χ 100 mm civil engineering concrete moulds. The friction angle between the cubes and the platens of the testing machine was varied by using soap and sandpaper. Different modes of failure were observed depending on the friction angle. Of interest is that significant loadshedding was recorded for some specimens which visually remained mostly intact. This highlights the difficulty of classifying pillars as failed or intact in underground stopes where spalling is observed. The laboratory models enabled a more precise calibration of the limit equilibrium model compared to previous attempts. Guidelines to assist with calibration of the model are given in the paper. The limit equilibrium model appears to be a useful approximation of the pillar failure as it could simulate the stress-strain behaviour of the laboratory models.

Simulating pillar reinforcement using a displacement discontinuity boundary element code

In this study we explore the use of a novel numerical modelling approach to study the effect of pillar reinforcement on pillar stability. Case studies in the literature indicate that tendons, strapping of the pillars, and shotcrete or thin spray-on liners are commonly used to reinforce pillars. No clear methodology exists to select the type of support or to design the capacity of the support required, however. This has led to ongoing collapses in some mines in spite of heavy support being used to reinforce unstable pillars. A limit equilibrium model with confinement on the edge of the pillar was used to simulate the interaction of the support with the failing pillar. The model correctly predicts that an increase in confinement will lead to a decrease in the extent of pillar failure. As the displacement discontinuity boundary element method allows for the efficient solution of large-scale bord-and-pillar layouts, the effect of pillar confinement can now be studied on a mine-wide scale. Accurate calibration of the limit equilibrium model is, however, required before this method can be used for the design of effective pillar support.

A study of backfill confinement to reinforce pillars in bord-and-pillar layouts

This study explores the use of backfill in hard rock bord-and-pillar mines to increase the pillar strength and extraction ratio at depth. The use of backfill will also minimize the requirement for tailings storage on surface and the risk of environmental damage. A literature survey indicated that backfill is extensively used in coal mines, but rarely in hard rock bord-and-pillar mines. To simulate the effect of backfill confinement on pillar strength, an extension of the limit equilibrium model is proposed. Numerical modelling of an actual platinum mine layout is used to illustrate the beneficial effect of backfill on pillar stability at greater depths. The magnitude of confinement exerted by the backfill on the pillar sidewalls is unknown, however, and this needs to be quantified using experimental backfill mining sections equipped with suitable instrumentation.

A study of the effect of pillar shape on pillar strength

Pillar strength is affected by pillar shape, but this has largely been ignored in past research studies. Bord-and-pillar layouts are typically designed using empirical strength equations developed for square pillars. Owing to the poor quality of pillar cutting, many hard-rock pillars have an irregular shape and it is not clear how this affects pillar strength. Furthermore, the strength of rectangular pillars in comparison with square pillars is also difficult to quantify. The 'perimeter rule' is widely adopted for rectangular pillars, but its applicability for pillars with irregular shapes has never been tested. We used numerical modelling in this study to investigate the effect of pillar shape on strength. An analytical limit equilibrium model of a square and a strip pillar also provided useful insights. For slender pillars, the strength of a long rib pillar is essentially similar to that of a square pillar. In contrast, for rib pillars with a large width to height ratio, there is a substantial increase in strength. The study found that the perimeter rule should not be used for irregularly shaped pillars. Displacement discontinuity modelling, using a limit equilibrium approach, is proposed as an alternative to determine the strength of these pillars.

Confidence and performance in different engineering tasks

To explore how different tasks inform decisions, geotechnicians were invited to predict the stability of a failed tailings dam. Participants included students, contractors and experienced consultants. The first task involved noting down stability concerns and making a stability prediction. Participants then prepared sketches, suggested material parameters and updated their predictions. The third task involved suggesting strength parameters for a limit equilibrium model, critiquing the model and making a final stability prediction. A 30 min time limit increased the chance of human error. Accurately assessing the failure mode at the note and sketch sections did not translate into correct predictions, despite high confidence in these tasks, particularly from experienced participants. The majority (80%) of final stability predictions were highly correlated with the analytical computer model, despite low confidence in this model, particularly from experienced participants. While this trend in confidence is perhaps expected, most participants, including experienced participants, failed to identify that failure was imminent and were confident in their final predictions. This was largely because many suggested drained strength parameters, rather than undrained strength parameters appropriate for the problem. This highlights the importance of intuitive tasks such as note taking and sketching to understand problems before building analytical models.

Identification of soil strength parameters for assessing stability of a sliding slope

The article presents the results of assessment of stability of the sliding slope, which was performed with the use methods of limit equilibrium and numerical modeling of the stress-strained state of the slope soils. It is obvious that usage of soil strength parameters c and φ without their clarification (identification) can lead to an overestimation of the value of the safety factor and thus slopes will be mistakenly classified as stable. The choice of the correct soil parameters is a key factor in the calculations in order to obtain calculated slip surfaces close to the actually recorded ones. The object of the study is an artificial landscape massif located on a slope. The site has the shape of an amphitheater in the form of stairs that occupy half of the entire area in the central part and sloping surfaces at the edges. The radius of the upper face is 42.20 m, the lower face is 21.95 m. The total number of steps is six. The absolute mark of the first step is 289.00 m, the mark of the bottom of the amphitheater is 280.00 m. During construction works related to the formation of a stepped landform of the landmass, a landslide happened on the territory of the object, therefore, during the calculation of the stability of the slope, the main task was to choose the correct values of c and φ so that the modeling results coincide with the data of field observations and on the basis of these data, it would be later on possible to make reasonable engineering decisions about measures to stabilize the slope and ensure its stability at the stage of operation. To assess stability of the slope, a series of calculations was performed in the initial (natural) state and in the current state with characteristics changed due to wetting. Soil parameters were identified on the basis of engineering and geological investigations. As a result of specifying the strength parameters, the coefficients of the stability were determined, which reflect the ultimate equilibrium of the slope, which corresponds to the real picture in the area. For numerical modeling, the initial stability coefficients had the following values: for 3 steps K=2.396; with complete excavation K=1.428. After adjusting c and φ, they decreased by 2.3 times and 1.4 times, respectively. A similar situation occurred for the assessment of slope stability using the limit equilibrium method. For three steps, the initial coefficient K=2.70 decreased by 2.5 times; with complete excavation, K=1.54 decreased by 1.5 times.

A bulking model to simulate stope convergence in deep tabular excavations

This paper investigates the use of a novel time-dependent bulking model in a displacement discontinuity code to simulate the convergence between the roof and the floor of deep tabular stopes. This study is important as the current layout design criteria that are based on elastic theory have become outdated. This approach ignores rock failure and fails to predict the actual magnitude and the time-dependent nature of the stope convergence. This has implications for support design and for estimating the stress acting on remnants in mature mines. A rock mass modulus reduction strategy, often used to circumvent this problem, reflects the expected lower effective rock modulus for near-stope fractured material, but must be applied for the entire rock mass region in the displacement discontinuity solution procedure for tabular layout problems. As an alternative, it is proposed in this paper that the rock bulking can be expressed as a function of the elastic closure by using a constitutive rule for the reef-normal compressive reaction stress that is expressed as a function of the compaction strain. The proposed model was tested using convergence data collected from a shaft pillar extraction in a deep gold mine. The bulking model, in conjunction with a limit equilibrium model to simulate the face crushing, seems capable of replicating the underground behaviour. As the stresses on remnants and pillars can be more accurately simulated, this approach holds promise of improving current design criteria. Calibration of the model nevertheless remains a challenge and a programme of routine convergence measurements will have to be implemented on the mines.

Stability Analysis of Antidip Bedding Rock Slopes with Very Low-Persistent Cross Joints Using a Limit-Equilibrium Model and Fracture Mechanics

Antidip bedding rock slopes always contain a developed set of joints that dip into the free surface; however, sometimes they have some gently inclined cross joints, and the persistence ratio (k) of these cross joints exerts important influences on the stability and failure modes. In this work, a limit-equilibrium model combined with a fracture mechanics (FM) method was established to predict the stability of antidip bedding rock slopes with very low-persistent cross joints. Rock layers that might undergo toppling failure were analyzed with the theories of Mode I fracture in FM; layers with the potential for shear–sliding failure were studied by solid mechanics and theories for Mode II fracture from FM. Then, a practical slope and a model test (reported in previous studies) were used to verify the feasibility of the proposed method. The calculated results were consistent with the actual state of the slope and the experimental result, which indicated that a limit-equilibrium model combined with an FM method could be used to predict the stability of such slopes. To further approve the rationality of the method, a contrastive analysis was conducted that used two other methods. The results showed that the predicted factors of safety (FS) of the proposed method were conservative and on the safe side. The proposed method could be used by engineers to evaluate and design such rock slopes.

A Limit Equilibrium Model to Simulate Time-Dependent Pillar Scaling in Hard Rock Bord and Pillar Mines

This study investigated the use of a limit equilibrium model to simulate the time-dependent scaling of hard rock pillars. In the manganese bord and pillar mines in South Africa, extensive scaling is observed for pillars characterised by a high joint density. It appears that the scaling occurs in a time-dependent fashion. Evidence for this is the ongoing deterioration of pillars in old areas, even after the pillars are reinforced with thin spray-on liners. Monitoring of selected pillars were conducted in an attempt to quantify the rate of time-dependent scaling. Contrary to expectations, almost no additional scaling was recorded for the pillars during a 3-month monitoring period. The scaling distance for pillars of different ages could be measured and it seems as if most of the scaling occurred soon after the pillars are formed. Only a limited amount of additional time-dependent scaling seems to occur after this. Numerical simulations of the time-dependent scaling were conducted using a displacement discontinuity code and a limit equilibrium constitutive model. The postulated exponential decay of the failed rock mass strength at the edges of the pillars resulted in simulated behaviour that is qualitatively similar to the underground observations. The results from this study are encouraging and the method can be used to investigate the long-term stability of bord and pillar excavations. Further work is required to improve on the calibration of the model and to better quantify the rate of scaling of the underground pillars.

Stability Evaluation of Sliding-Type Perilous Rock in Huangzangsi Hydrojunction Project Based on Natural Vibration Frequency

The instability of perilous rock is mostly manifested as sudden collapse and failure without obvious displacement characteristics. Therefore, it is difficult to achieve the purpose of monitoring and early warning by conventional displacement monitoring. But the existing stability monitoring indicators are mostly deformation, stress, and strain. There is a problem that the stability evaluation parameters are inconsistent with the monitoring parameters. Taking sliding-type perilous rock as the research object, the structural plane is assumed to be homogeneous and isotropic, and linear elastic deformation in the amplitude range. Based on the dynamic theory and limit equilibrium model, the quantitative relationship model can be established involving safety factors, natural vibration frequency, structural surface bonding area, elastic modulus, and mass. The remote laser vibrometer is used to monitor the natural vibration frequency of the sliding-type perilous rock on the slope of the Huangzangsi Hydrojunction, and the stability evaluation of the perilous rock is achieved based on the quantitative relationship model between the safety factor and the natural vibration frequency. In this way, the frequency of slipping perilous rock stability evaluation and safety factor can be monitored. The results are basically the same with the safety factor calculated by the limit equilibrium method, indicating that the method is correct and feasible. The research has high theoretical significance and practical value for the safety monitoring and advanced warning of sliding perilous rock.

A geotechnical evaluation of the Cumba Pit Slope Failure, Dominican Republic

The combined availability of topographical, geological, structural, hydrogeological and monitoring data is rapidly increasing. Technology and software advances allow the real time incorporation of this data across various software platforms. This paper describes the back-analysis of a 70 m high, pit slope failure of an open pit gold mine in the Dominican Republic, using data from aerial photogrammetry, ground-based synthetic aperture radar and 3D limit equilibrium and finite element modelling. This back-analysis process is considered leading practice with the latest technology. The (northern) side of the Cumba pit slumped along a non-daylighting plane that was identified after the failure event. Remedial investigations included review of geological data, major structures, rock mass constitutive models and groundwater conditions. Topographical and structural data acquired from aerial photogrammetry, pre- and post-failure event, was input into 3D models to replicate observed ground movement. 3D models of pit progression were compared with displacements recorded by ground-based synthetic aperture radar to calibrate model inputs and increase reliability of forward predictions. Such a technical review was completed in less than one week, and the review process implemented for the Cumba pit slope failure now forms the baseline approach for all future geotechnical analysis at the operating mine.

Tunnel face stability model for layered ground with confined aquifers

Confined aquifers with high fluid pressures pose severe risks for the stability of the face of earth-pressure-balanced shields. This study presents a method to compute the limit face pressures in the layered ground with confined aquifers. Seepage analyses based on the finite element method (FEM) are conducted to investigate the underground water flow patterns in the above ground due to shield tunneling. These analyses show: (1) the presence of confined aquifers enhances the hydraulic gradient around the tunnel face; (2) the hydraulic heads display distinct distribution patterns in the confined aquifers and adjacent aquitards; (3) the hydraulic gradients exhibit abrupt jumps at the aquifer-aquitard interface. These characteristics are in strong contrast to the distribution functions assumed in existing studies. Therefore, a new analytical flow model is proposed based on the hydraulic fields obtained from the seepage analyses. An approach to evaluate necessary face pressures is constructed by combining the flow model and a limit equilibrium model (LEM). By comparing against the results from drained FEM analyses under seepage flow, this paper shows that the proposed LEM can reasonably represent the influences of confined aquifers on the critical face pressures and the collapse mechanism of soils in front of the tunnel face.