Morgenstern Price Research Articles

A Generalized Limit Equilibrium-Based Platform Incorporating Simplified Bishop, Janbu and Morgenstern–Price Methods for Soil Slope Stability Problems

Limit equilibrium (LE) method is the most widely used method for slope stability analysis. Different methods based on the LE technique for the analysis of the stability of the slope have been developed. Some are based on satisfying the force equilibrium condition of the failing mass (Janbu’s method), while some focus on satisfying the moment equilibrium condition (Bishop’s method). Among these methods, the most accurate result is provided by the Morgenstern–Price method as it not only satisfies both moments as well as a force equilibrium condition but also considers the interslice shear forces (Vi) and interslice normal forces (Ei), which are neglected by most of the LE methods to avoid the condition of indeterminacy. To accommodate these forces, Morgenstern–Price (MP) gave a relation between the Vi and Ei which depends upon a scaling multiplier (λ). Thus, it becomes necessary to evaluate λ value along with the factor of safety (FS). There is barely any work discussing the detailed methodology of evaluation of λ along with FS. Method for obtaining λ along with FS have been developed and elaborated in details here. While calculating FS (MP method), evaluation of Ei is a must which is dependent upon the values of normal force at the base of each slice (Ni) and FS, which itself is dependent upon the value of Ei, making it a loop of interdependent variables. To avoid this interdependency of above stated variables, a separate formulation of Ei is given which reduces the calculations (run-time) involved. A VBA code-based platform has also been developed incorporating the generalized LE method, including Bishop’s, Janbu’s, and Morgenstern–Price methods which are represented in the form of flowcharts in this work.

A novel stability equation for the estimation of the factor of safety for homogeneous dry finite slopes

AbstractThis paper introduces a novel closed‐form equation (surrogate model) for approximating the Morgenstern–Price estimate of the factor of safety of homogeneous dry finite slopes with circular failure surfaces. Unlike typically used methods, the proposed equation does not require the definition of a critical failure surface, splitting the soil mass into slices, or the iterative reduction of soil resistance to the limit state. It can be easily programmed into calculators or computers and accurately determines the minimum factor of safety based on the shear strength parameters and slope geometry—meaning that it is ideally suited for integration into reliability calculations. The equation is determined parametrically for various soil parameters and slope geometry using the Morgenstern–Price method and compares favorably with conventional techniques, such as slope stability charts, limit equilibrium, and strength reduction methods (SRM). From a practical perspective, the proposed equation greatly simplifies the analysis of the slope stability as the creation of a numerical model is not required and is suited to use in the feasibility stage of a project, for example, for large linear infrastructure projects with differing slope geometries or in situations where a quick assessment is desired.

Efficient numerical implementation of limit equilibrium method for stability analysis of unsaturated soil slopes using Gaussian integral

Unsaturated soil is widely distributed around the world but less considered in design due to the absence of a convenient analysis method in practice. The Morgenstern–Price (MP) method incorporating the extended Mohr–Coulomb shear strength equation provides a reliable approach to evaluate slope stability in such conditions. However, this method is time-consuming due to the need for a tedious trial-and-error process in determining the scaling factor, which involves complex iterations during each trial. Furthermore, since the relatively complicated nature of unsaturated soil, a dense slice division is necessary to obtain reliable results, making the analysis even more cumbersome. In this paper, an improved MP method for unsaturated soil slope stability analysis is presented, which eliminates the need for a dense slice mesh by employing only a few strategically placed Gauss points along the slip surface. Moreover, the trial-and-error process for determining the scaling factor with the corresponding complex iterations is replaced by an efficient search algorithm with a more concise iteration process, resulting in a more convenient implementation of the proposed method. Extensive examples are provided to validate the effectiveness of the proposed improved MP method, indicating its potential as an accurate and efficient analysis method for unsaturated soil slopes in practical application and relative study involving repetitive analyses.

Slope Stability Evaluation and Geometrical Recommendation Using The Morgenstern Price Method

The study is located on PT Indocement Tunggal Prakarsa Tbk., which is still actively engaged in mining operations. The study aimed to determine the slope Safety Factor (SF) and offer suggestions for safe slope geometry for mining operations. Primary data collection through observation of lithology conditions, sample testing in the laboratory, scanline mapping, slope geometry measurements, and secondary data obtained from company inventories and related sources. The kinematic approach is used to assess the risk of landslides. This slope stability study uses boundary equilibrium based on the Morgenstern Price and Mohr-Coulomb failure criteria. Limestone is the majority of the rocks in the location. The kinematic analysis demonstrated that direct toppling and wedge toppling are the landslide potential at the research location. Using the non-circular boundary equilibrium approach, slope stability analysis was carried out on four Regions with dry and saturated groundwater conditions i.e, Regions A', B140, C120, and C135. The four Regions have stable slopes and are in good condition (SF values over 1.25). For the excavation to be carried out optimally, the recommendations for optimization of the overall slope geometry are given, namely, the height of the bench is 10 m, and the width of the ladder is 4 m. The slope angle is 80o, with SF value of 3.035 in dry conditions and an SF value of 2.021 in saturated conditions.

Deformation triggers and stability evolution of landslide from multiple observations

External causes like changes in reservoir level and intense rainfall can cause reservoir landslides. Exploring the factors that govern landslide deformation and analyzing its stability evolution is essential in mitigating the associated risks. The Sanzhouxi landslide, which has experienced ongoing movements and has been implemented a professional monitoring system, is chosen for analysis in this paper. A combination of geological survey and analysis of monitoring data is utilized to explore landslide deformation characteristics. A data mining method, grey relation analysis (GRA), is subsequently performed to determine the causes of landslide deformation. Furthermore, the stability of the Sanzhouxi landslide in response to reservoir level fluctuation and rainfall for each day over an entire year is assessed using the Morgenstern-Price (MP) approach in 2D GeoStudio software. Such a process illustrates clearly how the landslide stability alters with external triggers changing. The findings reveal that the landslide deforms variably in spatial and temporal. The reservoir level rising contributes to landslide deformation primarily, while rainfall has a secondary impact. The factor of safety (FS) of the Sanzhouxi landslide drops from 1.17 to 1.07 during high reservoir water level periods and remain the same or increase in other periods except for some transitory moments while decreasing only by about 2% under the effect of rainfall. The daily FS results validate the dominant influence of reservoir level fluctuation on the stability of the landslide. The comprehensive understanding of landslide movement based on deformation characteristics, triggering factor identification, and daily stability validation, contributes to realizing nearly real-time prediction and evaluating the risk due to slope movements in similar geological settings worldwide.

A Displacement-Dependent Limit-Equilibrium Slice Method for Slope Stability Analysis

Displacement and factor of safety are two crucial pieces of information about slope stability, but the traditional limit-equilibrium slice method has a significant defect; that is, it cannot reflect slope displacement. In this paper, a displacement-dependent limit-equilibrium slice method is proposed based on the soil shear constitutive model and Morgenstern–Price assumption of interslice forces to establish the relationship between the factor of safety and displacement. In addition to the overall factor of safety, point factors of safety and point displacements can also be obtained using the proposed method to characterize the displacement-dependent slope stability. The point factor of safety and shear displacement of each soil slice calculated using the proposed method are slightly higher and smaller than those calculated by the modified displacement-dependent Janbu method. The iterative process of the proposed method is simpler and more stable than that of the modified displacement-dependent Janbu method. Calculation results of four examples show that the point factors of safety at the upper and lower ends of the slide mass are generally smaller than those at the middle part of the slip body, and the shear displacement of the soil slice near the slope top is generally larger than those at the other positions. The proposed method associates the factors of safety with slope displacements, which allows us to quantitatively assess the slope stability immediately in practice by monitoring the displacements of some key points on it.

Stability Analysis of the Tailings Dam for the Purpose of Closing, Greening, and Ensuring Its Safety—Study Case

Tailings dams are special constructions that are part of a complex of works related to the installations for preparing mining masses. These constructions play a role in the mechanical treatment of wastewater and the safe storage of sterile material resulting from ore processing. The closing and greening of tailings dams is achieved taking into account the general stability of the pond and its related constructions, as well as the integration into the surrounding environment of the surfaces occupied by mining waste deposits (tailings dams). This study presents the results of the hydrogeotechnical and stability study carried out on the Gura Roşiei tailings dam location. This analysis aims to evaluate the stability degree of the three compartments that comprise the tailings deposit in order to carry out the closing and greening works of the tailings dam, and to conclude whether the idea of raising them by 1.5–2 m with tailings is feasible. This study was based on field observations, geotechnical drilling, and physical–chemical analyses of the collected samples. Due to the shallow depth, the drilling could not highlight a hydrostatic level except for the areas with excess humidity, areas represented by certain lenticular intercalations of sandy dust that yield water very slowly. These areas are not continuous and cannot define a reference hydrostatic level. All situations were analyzed by four different methods that satisfy the static equilibrium of forces or moments (Bishop and Janbu) or simultaneously of forces and moments (Spencer and Morgenstern–Price). From the point of view of the stability calculations performed in the hypothesis in which the three ponds become active for the storage of tailings, assuming a corresponding piezometric level, the resulting safety factors are relatively close to the standard values (Fs ≥ 1.4) for the static analysis, and in seismic conditions, they are at the limit of equilibrium. The NE slope of pond No. 2 shows values below the standard safety limit for this type of work. Moreover, tailings dam No. 3 presented from the calculations as being totally inadequate for the elevation. From the obtained results, it was found that the location formed by the three compartments that comprise the Gura Roşiei tailings dam presents major disadvantages for a future storage of flotation tailings, being at the same time an imminent danger to the environment. Due to its reduced capacity to release water from its pores, the settled material is still in a saturated state, and it is assumed that the foundation land, comprised of the terrace deposits of the Abrudel River, is clogged at the interface with the settled material and unable to naturally drain the excess moisture from the dam’s body.

An efficient method for computing slope reliability calculation based on rigorous limit equilibrium

Abstract Traditional rigorous limit equilibrium methods satisfy all equilibrium conditions and usually have high accuracy, however, which are less efficient for slope reliability analysis. The main reason is that the limit state functions are highly nonlinear implicit functions of safety factor. Complex numerical iterations are required, which may sometimes lead to computational convergence problems. A new method for computing slope reliability calculation with high efficiency and accuracy was proposed. This method was based on the rigorous limit equilibrium method by modifying normal stresses over the slip surface. The critical horizontal acceleration factor K c {K}_{c} , which can be expressed explicitly, was used to replace the implicit safety factor as a representation of slope stability. The difference between K c {K}_{c} and the known value K c 0 {K}_{c0} was used as the limit state function. Two slope examples were analyzed. The results showed that the calculation results of this method were in good agreement with those of the traditional Morgenstern–Price limit equilibrium method, but the computational efficiency was significantly improved. When this method was combined with the subset simulation method, the calculation time was only a few seconds. Therefore, this method can be used for rapid calculation of slope reliability.

Using the Morgenstern–Price Method and Cloud Theory to Invert the Shear Strength Index of Tailings Dams and Reveal the Coupling Deformation and Failure Law under Extreme Rainfall

It is difficult to obtain reliable shear strength parameters for the stability analysis and evaluation of tailings dams in an unstable state. In this study, the sensitivity of the shear strength index to the safety factor of a tailings dam was evaluated. The cohesion C range of a tailings earth rock dam in an unstable state is determined by the safety factor, and the Morgenstern–Price method is used for inversion. During parameter inversion, uncertainty reasoning is established based on cloud theory, which overcomes the problem that the fuzziness and randomness of the quantitative cohesion value are transferred to the qualitative concept of the safety factor. The results show that the change in cohesion C has a greater influence on the safety factor Fs of the tailings dam, and the value of parameter inversion is 8.6901 kPa. The deformation and failure of tailings dams under extreme rainfall conditions are analyzed by using the modified cohesion C value. The dam toe becomes the main response area of plastic deformation and slowly expands to the interior, showing creep deformation. The displacement field gradually transfers from the accumulated tailings to the tailings dam with the flow direction, causing erosion damage. This study provides a new idea and method for parameter inversion of the shear strength index of tailings dams and provides a reference for the disaster prediction and prevention of tailings dams subjected to extreme rainfall.

Optimum Design of Unsaturated Finite Clayey Slopes Using Second-Order Reliability Method

This paper discusses the second-order reliability-based design optimization (SORBDO) of unsaturated finite slopes. An efficient search algorithm is proposed to locate the most probable slip surfaces using the Morgenstern–Price method of slices. The variability of fitting parameters of the soil water characteristic curve (SWCC) and shear strength parameters have been taken into account for evaluating the reliability of unsaturated finite slopes. Six unsaturated shear strength (USS) models are considered for the analysis of unsaturated soils. The most conservative model for the optimum design of finite slopes is recommended. Moreover, the results indicate that the mean and standard deviation of the fitting parameters of SWCC have a significant effect on the critical centers, critical slip surfaces, and reliability index of the unsaturated finite slopes. The results demonstrate a significant change in the critical centers associated with critical slip surfaces due to the change in the mean and coefficient of variation of the fitting parameters of SWCC. Statistical analysis of the experimental data shows that a positive correlation exists between fitting parameters of SWCC related to air entry value (af) and residual water content (mf). Negative correlations are found between the fitting parameters of SWCC related to air entry value (af) and slope (nf). Similarly, the SWCC fitting parameters related to residual water content (mf) and slope (nf) have negative correlations for clayey soils. It is observed that the correlated value of the reliability index is overestimated with reference to the uncorrelated value for clayey soil slopes. The proposed design charts are beneficial for engineers and practitioners for the sustainable design of unsaturated finite slopes. Overall, the developed rigorous SORBDO framework offers a more rational approach for assessing critical failure surfaces and critical centers, considering the interdependency of random variables.

Application of interval field method to the stability analysis of slopes in presence of uncertainties

Spatial uncertainty of soil parameters has a significant impact on the analysis of slope stability. Interval field analysis is emerging as a complementary tool of the conventional random field method that can take spatial uncertainty into account, which, however, has not been investigated in slope stability analysis. The present paper proposes a new method, named the interval field limit equilibrium method (IFLEM), for assessing the stability of slope in the presence of the interval field. In this method, the modified exponential function is introduced to characterize the spatial uncertainty of the interval field and the Karhunen–Loève-like decomposition is employed to generate the interval field. Then, in a single calculation, the deterministic slope stability analyzed by the Morgenstern–Price approach is implemented in order to estimate the safety factor. Subsequently, the upper and lower bounds of the interval of safety factor are efficiently evaluated by a kind of surrogate-assisted global optimization algorithms, such as Bayesian global optimization used in this study. Finally, the effectiveness of the proposed method is verified by three numerical examples. The results indicate that the proposed method can provide reasonable accuracy and efficiency, which is potentially applicable to a number of geotechnical systems.

Minimum Safety Factor Evaluation of Slopes Using Hybrid Chaotic Sand Cat and Pattern Search Approach

This study developed an efficient evolutionary hybrid optimization technique based on chaotic sand cat optimization (CSCO) and pattern search (PS) for the evaluation of the minimum safety factor of earth slopes under static and earthquake loading conditions. To improve the sand cat optimization approach’s exploration ability, while also avoiding premature convergence, the chaotic sequence was implemented. The proposed hybrid algorithm (CSCPS) benefits from the effective global search ability of the chaotic sand cat optimization, as well as the powerful local search capability of the pattern search method. The suggested CSCPS algorithm’s efficiency was confirmed by using mathematical test functions, and its findings were compared with standard SCO, as well as some efficient optimization techniques. Then the CSCPS was applied for the calculation of the minimum safety factors of the earth slope exposed to both static and seismic loads, and the objective function was modeled based on the Morgenstern–Price limit equilibrium method, along with the pseudo-static approach. The CSCPS’s efficacy for the evaluation of the minimum safety factor of slopes was investigated by considering two case studies from the literature. The numerical experiments demonstrate that the new algorithm could generate better optimal solutions via calculating lower values of safety factors by up to 10% compared with some other methods in the literature. Furthermore, the results show that, through an increase in the acceleration coefficient to 0.1 and 0.2, the factor of safety decreased by 19% and 32%, respectively.

An improved morgenstern-price method using gaussian quadrature

The Morgenstern-Price (MP) method is an effective solution to compute the factor of safety (FOS) of a slope against sliding that has been used extensively for decades. In the MP method, the soil mass above the slip surface needs to be tediously divided into vertical slices to compute the force and moment integrals. The proposed research improves the computational efficiency of the MP method by employing the Gaussian quadrature for the integrals, where the soil mass division into a significant number of vertical slices is improved by aligning a few Gauss points along the slip surface. Consequently, the computational efficiency can be dramatically improved, resulting in a reduction in calculation efforts. Moreover, this research further refines the iteration scheme to satisfy the equilibrium conditions for the FOS using a robustly stable numerical algorithm with two initial scaling factors. The critical slip surface was determined by the simulated annealing algorithm. Finally, the proposed method was thoroughly validated via a series of benchmark examples. This research could improve the traditional MP method by enhancing its robustness and efficiency and provide an informative technique for force integrals that may be motivated by other problems in geotechnical engineering.

Determining the Critical Slip Surface of Slope by Vector Sum Method Based on Strength Reduction Definition

Slope-stability assessment involves the location of the critical slip surface and the corresponding factor of safety (FOS). Considering force-vector characteristics, the vector sum method was further studied based on the strength reduction definition of the FOS and stress field of slope, and the FOS can be calculated directly by the force limit equilibrium equation on the global sliding direction of the slope. For the sliding direction, it is rigidly proved to be determined only by the sliding shear stress along the slip surface based on the principle of minimum potential energy. Then, two examples with fixed slip surfaces were analyzed and the results were compared with the rigorous Morgenstern–Price method. Finally, two examples from the literature are investigated for searching the critical slip surface using the proposed method and a real-coded genetic algorithm. The calculated results revealed the rationality of the proposed method for both fixed slip surfaces and critical slip surfaces from the literatures in slope stability assessment. This proposed method provides a new path to assess the slope stability, which is worth to be further studied in practical engineering.

Analytical Reliability Evaluation Framework of Three-Dimensional Engineering Slopes

An analytical three-dimensional slope reliability evaluation framework was developed in this work independent of use of numerical simulations. The slope stability analysis was necessarily carried out by utilizing an extended three-dimensional Morgenstern–Price method, which was characterized by analytical formulations and competitive computational efficiency. Incorporation of the presented stability analysis method into response surface methodology led to an effective slope reliability evaluation framework. The applicability and superiority of this framework was examined and validated using a real complicated landslide case reported in practice, and a hypothetical slope example widely adopted in the literature. The impact of correlation coefficients and probability distribution patterns on the slope reliability assessment results was further addressed to derive additional benefits of this framework.

Seismic Analysis of Earth Slope Using a Novel Sequential Hybrid Optimization Algorithm

One of the most important topics in geotechnical engineering is seismic analysis of the earth slope. In this study, a pseudo-static limit equilibrium approach is applied for the slope stability evaluation under earthquake loading based on the Morgenstern–Price method for the general shape of the slip surface. In this approach, the minimum factor of safety corresponding to the critical failure surface should be investigated and it is a complex optimization problem. This paper proposed an effective sequential hybrid optimization algorithm based on the tunicate swarm algorithm (TSA) and pattern search (PS) for seismic slope stability analysis. The proposed method employs the global search ability of TSA and the local search ability of PS. The performance of the new CTSA-PS algorithm is investigated using a set of benchmark test functions and the results are compared with the standard TSA and some other methods from the literature. In addition, two case studies from the literature are considered to evaluate the efficiency of the proposed CTSA-PS for seismic slope stability analysis. The numerical investigations show that the new approach may provide better optimal solutions and outperform previous methods.

Stability analysis of mine overburden dump and improvement by soil nailing

The conventional method for carrying out stability analysis of dump involving Limit Equilibrium Method (LEM), when compared with the strength reduction method (SRM) using Finite Element Method (FEM) approach, resulted in overprediction of the factor of safety (FoS). In the present study, slope stability analysis of external overburden dump in Sonepur Bazari opencast mine has been carried out. Morgenstern Price method using constant inter-slice force function conducts the analysis of LEM using Geo-studio Slope/w, and the FEM analysis is carried out using the PLAXIS 2D 2016. The study is carried out to assess the effect of the dump’s friction angle, slope angle, height, and bench configuration. Further, the analysis considering the soil nailing is also conducted by optimizing the number of nails, the distance between nails, length of the nails, nail inclination angle. The stability analysis found that the dump is in a critical state (FoS = 0.92) with a circular failure surface. The optimum slope angle is found to be 30˚. It is observed that LEM cannot detect the increased localized stress concentration at the bench edge. Reinforcement of the dump using the soil nailing technique has affected the dump stability substantially, with an increase of 11% in the FoS; however, the effect of nail orientation on dump stability was less pronounced. Further bench provision to the dump slope increases the FoS by 27%, 21% and 19%, considering the total height of 60 m, 80 m and 100 m, respectively.

Field Treatment of Storage Sludge and Stability Analysis of Overlying Municipal Waste Landfilling

Storage sludge has high water content and low shear strength, which limits the capacity expansion of overlying municipal landfilling. Few studies have addressed the field treatment of large amounts of storage sludge due to the variability of the depth of geotechnical property. This paper proposes a stratified treatment method for storage sludge, based on the in situ characterization of layered sedimentary patterns of the storage sludge acquired from the Qizishan landfill in China. Additionally, the stability of the landfilling above the sludge pond is analyzed using the Morgenstern–Price and limit equilibrium slice method, which considers the layered strength properties of solidified sludge. The treated sludge has a significant decrease in average water content from 1398% to 88% and an increase in average cohesion to 23.52 kPa. The high content of clay particles, low amount of solidification products, and high water content together result in the high sensitivity to the water content of the strength of deep solidified sludge. For a 40-m high waste body, stability analysis suggests a sliding surface across the raw sludge pond, while the critical surface remains outside the treated sludge pond and the safety factor is increased from 0.934 to 1.464. The validated stratified treatment provides valuable references for the treatment of deep sludge.

Optimal design of the high geogrid-reinforced slope at the airfield of an airport

Geogrid reinforced soil slope is a new type of slope supporting structure, which is of great significance for improving the slope stability, saving construction land and preserving the ecological environment. In order to provide the optimal design solution of the reinforced soil slope for high fill slopes, a case study is conducted on the No.6 slope at the northwest corner of the airport runway. Three design schemes with different slope ratios are carried out firstly based on the geological conditions of the slope and the field situation of the high fill slope. Secondly, the simplified Bishop method, Spencer wedge method and Morgenstern price method are used to calculate the stability factors of the reinforced slopes under the natural, rainstorm and earthquake conditions. Finally, the deformation characteristics of the slope and tensile force in reinforcements are investigated based on the finite element method under the natural conditions. The results show that all the proposed design schemes can meet the requirements of slope stability under different working conditions. A zigzag distribution of the tensile force in reinforcements in the multi-stage reinforced soil slope is observed along the slope height, and the maximum axial force of reinforcement increases abruptly at the toe of each slope. Compared with the design scheme of the gentle reinforced soil slope with the slope ratio of 1∶1.5, the reinforced retaining wall with the slope ratio of 1∶0.25 shows obvious advantages in reducing the slope height, consumption of geosynthetics, filling and excavation amount as well as slope protection area. The key issues including the slope stability, construction cost and duration are comprehensively considered, and the design scheme of the reinforced earth wall is reasonable.

Factor of safety of pile-stabilised slopes: an algorithm incorporating soil-arching effect

The current practice of factor of safety computation of pile-stabilised slopes depends substantially on either the extension of methods for normal slopes with the pile-contribution introduced as an additional term or finite-element modelling. Extending conventional methods to analyse pile-stabilised slopes fails to capture the key mechanisms like soil arching, gradual transfer of resistance offered by the pile through the slope, change in slip surface due to introduction of piles and so on. In this paper, a new algorithm is proposed to compute the factor of safety of pile-stabilised slopes, in which the Morgenstern–Price method is modified to incorporate the effect of a pile. The resistance offered by the pile is obtained using a pressure-based method which considers the soil-arching effect in horizontal and vertical directions. The gradual propagation of this resistance offered by the pile is incorporated through an iterative procedure, unlike the conventional methods. The algorithm considers the change in slip surface due to the introduction of the pile. The factor of safety values computed using the proposed algorithm were found to be in close agreement with that obtained from finite-element modelling. Moreover, a simple technique to determine the optimal location of stabilising piles and pile-spacing is also presented.