Critical Failure Surface Research Articles

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.

Influência das superfícies sub-críticas no comprimento de reforços em estruturas de solos reforçados com geotêxteis ou geogrelhas

Design of steep slopes and retaining walls, reinforced with geotextile or geogrids, requires internal stability checks usually referred to a critical failure surface, which determines the amount of reinforcement required. In determining the length of the reinforcement layers the position of the critical surface and also of the sub-critical surfaces must be considered. In relation to these, are verified the anchorage lengths required to ensure the pullout resistance of the reinforcements. This paper presents a study based on limit equilibrium analysis, with bi-linear failure surface, to determine the amount and length of the reinforcement layers required. The model allows replacing Jewell charts by algebraic equations and iterative calculation processes. The results obtained for the minimum length of reinforcements agree with good accuracy for effective friction angles greater than or equal to 30° and indicate that shorter lengths can be used for materials with lower effective friction angle. Expressions for the calculation of the anchorage length are also presented for the three possible cases: anchorage in a section under the projection, part under the projection and beyond the horizontal projection of the slope face.

INFLUENCE OF SURFACE CRACKS ON THE STABILITY OF CRACKED SOIL SLOPE

The slope stability is a major concern to geotechnical engineers. Traditional methods of slope stability analysis have potentially ignored the influence of surface cracks. It is also known that seasonal rainfall and seepage through crack are closely related with slope failure. First, surface cracks provide special flow channels which increase the soil permeability and decrease the soil strength. Second, water-filled cracks apply an additional active force on the slope. Finally, cracks can create a part of the critical failure surface that has no shear strength. The objective of this paper is to investigate the influence of existing cracks on the stability of a cracked soil slope in different state. The effects of crack depth, slope angle and water-filled cracks on the stability of the cracked slope are explored. The analysis was conducted using the computer modelling programs Optum G2 to analysis of slope factor of safety. The results show that with increasing of slope angle the factor of safety decreases and this problem is significant in the slope with water filled cracks. Also, Factor of safety for all of slope angles in Dry and water filled cracks states with increasing the crack depth, decrease significantly.

Numerical estimate of critical failure surface of slope by ordinary state-based peridynamic plastic model

In this paper, two-dimensional ordinary state-based peridynamic (OSB-PD) plastic model is coupled mechanically with the Drucker-Prager (D-P) criterion, aiming at numerically reproducing localized deformation and locating the critical failure surface of slopes. A non-physical (or negative) incremental plastic energy can be avoided under extreme non-uniform deformation in this model. The strength reduction method is adopted to estimate the critical failure surface and factor of safety. For PD strength reduction method, three evaluation criteria are introduced to estimate the critical damage state of slopes. Two numerical simulations including a classical slope model and centrifuge tests of sand slopes are performed by the proposed PD method. The critical failure surfaces of slopes are obtained by PD and compared with the results obtained by (i) the finite element method (FEM), (ii) the simplified Bishop method, and (iii) the centrifuge tests. The numerical, analytical and experimental results have collectively verified the proposed PD method.

Study on Seismic Amplification Effect and Active Earth Pressure of Retaining Wall Based on Pseudo-Dynamic Method and Numerical Simulation

Abstract In this paper, the finite element numerical analysis verified that the critical failure surface of the backfill under active conditions was curved, and it was consistent with the composite form of a logarithmic spiral and a straight line assumed by the study based on the pseudo-dynamic method. On the basis of the failure surface of the backfill as a curved surface, the seismic active earth pressure on the retaining wall was calculated by the pseudo-dynamic method and the finite element method respectively. Results showed that in the finite element analysis, the acceleration amplitude of the backfill had an obvious amplification effect along the depth of the retaining wall, and this amplification effect was related to many factors. Under the action of low-frequency seismic load, linear amplification effect occurred, while nonlinear amplification effect occurred under the action of high-frequency seismic load. The amplification factors increased with the increase of retaining wall height. According to the characteristics of the amplification effect, after modifying the input seismic acceleration in the pseudo-dynamic method, the distribution of seismic active earth pressure on the retaining wall was close to that obtained by finite element analysis.

Stability and Deformation Responses of Earth Slopes Subjected to Multiple Natural and Manmade Hazards

Multiple hazards such as rainfall, earthquake, and excavation may occur simultaneously, and the stability (factor of safety) and deformation (movement) behaviors must be investigated to prevent loss of life and properties. In this study, the stability and deformation behaviors of earth slopes subjected to the individual and simultaneous occurrence of these events are investigated using a coupled Geotechnical–Hydrological finite-element method (FEM). First, the finite-element model was validated against a case history and a widely used limit-equilibrium method (LEM) considering site-specific soil properties and rainfall records. The shape and the size of the critical failure surface obtained from the field investigation coincided well with those calculated using FEM and LEM. Also, the predicted onset of slope failure (after eight days of rainfall) coincided well with the real-time occurrence of the slope failure. The validated finite-element model was then used to predict the stability and deformation responses of earth slopes subjected to several possible scenarios, including individual and simultaneous occurrences of these events. The rainfall–excavation scenario indicated that the slope would have failed two days before the recorded time if the slope had a vertical cut width of 4.25 m. The rainfall–earthquake hazard scenario caused the slope to move horizontally by 799 mm. The parametric study conducted by varying the friction angle of the slope soil by ± one standard deviation (STD) with 10% covariance showed that the slope would have failed one day later and two days prior to the mean friction angle, respectively, when subjected to rainfall. Also, varying the friction angle by +1 STD and −1 STD failed the slope for 0.5 m above and 0.4 m below the cut width for the slope with mean friction angle when subjected to excavation, respectively. Further, the percent increase in the horizontal movement of the slope soil with friction angle by +1 STD and −1 STD compared to mean friction angle was 56% and 18%, respectively, when subjected to earthquake.

A log-spiral limit equilibrium analysis for passive earth pressure under the effect of unsaturated seepage conditions

In the current state of the study regarding passive earth pressure, the majority of literature was limited in scope to dry or saturated cases. However, most soils in practical engineering are unsaturated. Based on the log-spiral failure mechanism, this paper develops a limit equilibrium (LE) analysis to evaluate the passive earth pressure under the effect of steady unsaturated seepage conditions. For assessing the effect of suction stress on passive earth pressure, a suction stress-based effective stress representation is incorporated into a limit equilibrium analysis. The analytical formula of passive earth pressure coefficient is derived using the moment equilibrium equation for the backfill defined by a log-spiral slip surface. An optimization procedure is developed to search for the minimum passive earth pressure coefficient and the critical failure surface via MATLAB. A parametric analysis is performed to investigate the effects of some key parameters on passive thrust. Besides, examples of two different cases are given to verify the effectiveness of theoretical results. Results indicate that suction stress can increase the magnitude of the passive earth pressure coefficient and the suction-induced effect highly depends on the seepage conditions and soil types.

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.

Simulating the entire progressive failure process of rock slopes using the combined finite-discrete element method

This paper presents a novel approach (Y-slope) to simulate entire slope failure processes, from initiation, transport to deposition. The algorithm is implemented in a combined finite-discrete element method code. Absorbing boundary conditions are implemented to improve computational efficiency for the initial stress state equilibrium. Strength reduction methods, considering both tensile and shear failure modes, are implemented to evaluate the slope stability, where the safety factor and critical failure surface are automatically obtained. The energy dissipation mechanism, due to blocks’ friction and collision, is incorporated to accurately simulate the block kinematics during the post-failure stage. The accuracy and robustness of Y-slope are validated by numerical tests, and the failure mechanism and failure progress of a homogeneous and jointed rock slope are presented. Results indicate that Y-slope can not only evaluate the slope stability state (e.g., safety factor and critical failure surface), but also simulate the entire failure process (e.g., slope deformation, failure surface evolution, block transport and deposition). In addition, the critical role of existing discontinuities on the slope stability and failure mechanism are also highlighted. This work proposes a promising tool in understanding the failure mechanism and assessing the potential risk by predicting the entire failure process of rock slopes.

Evaluation of maximum safe-slope angle using the kinematical element method and its application

The determination of the maximum safe-slope angle with a planned slope height is a critical issue in slope design and construction. In this study, to overcome the drawbacks of the rule of thumb and trial-and-error methods, the optimization of slope angle is treated as a root-finding problem. The kinematical element method (KEM) within a rigorous theoretical framework is used to solve this problem using the modified false-position method. The critical noncircular-failure surface and the associated minimum factor of safety are located using the KEM. A rock slope with planar failure is given for demonstrating the validity of this method. The result shows that the maximum safe-slope angle obtained using the KEM is consistent with the analytical solution. The method proposed in this study has a satisfactory convergence speed. In addition, a bench-shape fill slope in an iron and steel base is used as a case study. The maximum safe-bench face angle of the fill slope, under self-weight condition, is 41.43°, and it decreased by 17% due to seismic loading. Finally, the effect of the slope height on the maximum safe-slope angle is analyzed, and strong correlations that display an exponential function are found. The critical failure surface associated with the maximum safe-slope angle becomes deeper as the slope height increases.

Computational tools for the analysis of circular failure of rock slopes

Among other types of failure in slopes created by excavation or filling, circular (also referred to as rotational) type of failure plays an important role in the design and stability analysis of high rock slopes, such as those encountered in deep open pit mines. When a rock slope of significant height involves predominantly one type of jointed rock mass, the scale of the problem is frequently such that the rock mass (intact rock and rock discontinuities) can be regarded as a homogeneous isotropic material. In such case, if the rock slope is unstable and collapse occurs, circular (or rotational) failure as in failed slopes in soils is also observed for the rock slope. One main goal in practical design of deep open pit mines is to determine safe inclination angles for the pit faces to be excavated to avoid failure. Computational tools for quick assessment of the stability conditions, in particular determination of factor of safety and location of the critical circular failure surface, are useful during the preliminary stages of open pit design. Among others, such tools allow the engineer to gain insight in the role that the different variables controlling the problem have in defining the stability conditions for the rock slope. This paper provides first an overview of typical numerical techniques used to assess the stability of high rock slopes. For the case of slopes in rock masses that are assumed to obey the Mohr-Coulomb or the Hoek-Brown failure criteria, the paper discusses how application of transformation rules leads the criteria to be re-casted in dimensionless form, so when these are applied to the solution of slope problems, the results (factor of safety and coordinates that define the location of the critical circular failure surface) can be summarized in dimensionless compact forms. The paper presents then computational tools in the form of dimensionless charts and computer spreadsheets for stability analysis. These are developed from numerical limit equilibrium models that use the mentioned transformation rules. Among others, the computational tools allow the concept of mechanical similarity of rock slopes to be naturally introduced. Finally, the paper presents various rock slope examples that illustrate the use of the computational tools. In these examples, results from limit equilibrium models are also compared with results obtained with full numerical analysis, in particular with finite difference models implementing the shear strength reduction technique.

Landslide Susceptibility Assessment Using the Scoops3D Model in a Tropical Mountainous Terrain

Many physically-based distributed models study the landslide occurrence using an infinite slope stability analysis, simulating a planar failure, which is not usually applicable to rotational failures and deep landslides. Recently, some three-dimensional distributed physically-based models have been developed that have been applied in different parts of the world. In this research, the Scoops3D model is implemented for a landslide susceptibility analysis in a tropical mountainous terrain of the Colombian Andes (Medellín, Colombia). In addition to identifying the areas susceptible to the occurrence of rotational landslides, the results of the safety factor are analyzed with the areas of associated critical failure surfaces to provide an interpretation and explanation of the simulation results. This is to have a better understanding of how the model works and to facilitate its implementation in landslide hazard assessment. The Scoops3D physicallybased model can be a very useful tool for mass movement risk management projects.

Numerical Study of the Behavior of Back-to-Back Mechanically Stabilized Earth Walls

Back-to-back mechanically stabilized earth (MSE) walls can sustain significant loadings and deformations due to the interaction mechanisms which occur between the backfill material and reinforcement elements. These walls are commonly used in embankments approaching bridges, ramps, and railways. The performance of a reinforced wall depends on numerous factors, including those defining the soil, the reinforcement, and the soil/reinforcement interaction behavior. The focus of this study is to investigate the behavior of back-to-back mechanically stabilized earth walls considering synthetic and metallic strips. A two-dimensional finite difference numerical modeling is considered. The role of the soil friction angle, the distance of the reinforcement elements, the walls’ width to height ratio, and the quality of the soil material are investigated in a parametric study. Their effects on the critical failure surface, shear displacements, wall displacements, and tensile forces on the reinforcements are presented. The interaction between back-to-back reinforced walls strongly depends on the distance between walls and modifies the critical failure surface location.

Stability analysis of anti-dip bedding rock slopes using a limit equilibrium model combined with bi-directional evolutionary structural optimization (BESO) method

Anti-dip bedding rock slope (ABRS) is a very common type of rock slopes in nature and engineering construction. However, the stability assessment of ABRSs is still a hard problem for geotechnical engineers due to the difficulty in finding the critical failure surface. In this work, a new method called LEM-BESO was proposed to tackle this problem based on limit equilibrium theory and bi-directional evolutionary structural optimization method. Two model tests reported in previous studies were employed to verify the correctness of the LEM-BESO method. Moreover, a comparative study of the LEM-BESO method and the genetic algorithm was also conducted. The highly consistent results demonstrate that using the LEM-BESO method can accurately predict the failure surface of the slope. Then, the LEM-BESO method was extended to study the influence of two key parameters (i.e. friction angle of the joints and tensile strength of the rock layers) on the failure mechanism of anti-dip bedding rock slopes. The results show that these two parameters have a significant influence on the stability of the slope. However, the failure surface is insensitive to the tensile strength of the rock layers. The LEM-BESO method provides an effective means for the stability evaluation and engineering design of ABRSs.

Improved Vanmarcke analytical model for 3D slope reliability analysis

This paper presents an improved algorithm for evaluating the reliability of three-dimensional soil slopes based on the Vanmarcke model (VM). In the development of the improved Vanmarcke model (IVM), the sliding mass is assumed to be a portion of a cylinder bounded by two ellipsoidal end sections. The soil spatial variability of both the cylindrical and ellipsoidal sections is considered. A set of equations is derived to determine critical failure surfaces and compute their reliability indices. Numerical studies are performed to examine the IVM. The results indicate that (1) the IVM predicts a similar critical failure length compared with that computed by the VM. However, a lower reliability index is obtained by the former. (2) The relative difference between the reliability indices computed by the two analytical models can reach 20% for practical situations. The horizontal scale of fluctuation (SOF) of soil properties is the main factor affecting the relative difference. (3) Compared to the random finite element method, the IVM still gives unconservative analysis results, especially at low horizontal SOFs. This is due to the prediction of smaller critical failure lengths by the IVM for small horizontal SOFs and some retained assumptions of the VM when developing the IVM.

Evaluation of the Instability Risk of the Dam Slopes Simulated with Monte Carlo method (Case Study: Alborz Dam)

Embankment dams are one of the most important geotechnical structures that their failures can lead to disastrous damages. One of the main causes of dam failure is its slope instability. Slope Stability analysis has traditionally been performed using the deterministic approaches. These approaches show the safety of slope only with factor of safety that this factor cannot take into account the uncertainty in soil parameters. Hence, to investigate the impact of uncertainties in soil parameters on slope stability, probabilistic analysis by Monte Carlo Simulation (MCS) method was used in this research. MCS method is a computational algorithm that uses random sampling to compute the results. This method studies the probability of slope failure using the distribution function of soil parameters. Stability analysis of upstream and downstream slopes of Alborz dam in all different design modes was done in both static and quasi-static condition. Probability of failure and reliability index were investigated for critical failure surfaces. Based on the reliability index obtained in different conditions, it can be said that the downstream and upstream slope of the Alborz dam is stable. The results show that although the factor of safety for upstream slope in the state of earthquake loading was enough, but the results derived from probabilistic analysis indicate that the factor of safety is not adequate. Also the upstream slope of the Alborz dam is unstable under high and uncontrolled explosions conditions in steady seepage from different levels under quasi-static terms.

Rock slope stability charts based on limit equilibrium method incorporating Generalized Hoek–Brown strength criterion for static and seismic conditions

Stability charts present a graphical approach for the estimation of factor of safety (FOS) of a finite slope with uniform soil properties which are routinely used by geotechnical engineers. They provide a way for quick and efficient determination of stability of slopes sidestepping the need for carrying out actual analysis. In the present work, authors have developed stability charts of rock mass to determine global minimum FOS for rock slope stability problems. The rock mass is modelled by incorporating generalized Hoek–Brown strength criterion proposed by Hoek et al. (Hoek–Brown criterion—2002 edition, 2002). Limit equilibrium technique based Morgenstern and Price (Geotechnique 15:79–93, 1965) method is used to determine the value of factor of safety of the rock slope against failure. An evolutionary optimization method i.e. Particle swarm optimization (PSO) is used to search for the minimum FOS values and the associated critical failure surface out of all possible slip surfaces. A MATLAB code has been developed for this purpose. The charts have been developed for the entire range of Geological Strength Index (GSI) values ranging from 0 to 100 for intact rock mass characterized by the disturbance factor D = 0 for both static and seismic loading conditions. The seismic slope analysis is performed by an equivalent static method using a horizontal seismic coefficient (kh) with values ranging from 0.10 to 0.30. Stability charts contain stability numbers which are inversely proportional to FOS value of the slope. In the present work, stability numbers are plotted for different inclination angles ( $$\beta$$ ) equal to 15°, 30°,45°, 60° and 75+°, respectively, for both static and seismic conditions considering the variations of other material parameters such as $$m_{i}$$ and $${\text{GSI}}$$ . Stability numbers are observed to increase with an increase in angle of inclination ( $$\beta$$ ) of the slope and correspondingly FOS values follow a decreasing trend. However, stability numbers follow a decreasing pattern with increasing values of material parameters $$m_{i}$$ and $${\text{GSI}}$$ designating improved rock quality with corresponding increase in FOS values of the rock slope.

Efficient computational system reliability analysis of reinforced soil-retaining structures under seismic conditions including the effect of simulated noise

This article presents a computational reliability analysis of reinforced soil-retaining structures (RSRS) under seismic conditions. The internal stability of RSRS is evaluated using the horizontal slice method (HSM) with modified pseudo-dynamic seismic forces. Two different failure modes of RSRS are identified and their reliability indices are computed using the first-order reliability method (FORM). The critical probabilistic failure surface is identified using a three-tier optimization scheme. Reliability index of the system is computed by considering the modes of failure to be connected in series. The tension mode is found to be the most critical mode of failure. The present study identifies that the wall height (H), shear wave velocity of the soil (Vs), and predominant frequency of the input motion (ω) govern the response of RSRS. Reliability indices depend on a parameter termed as the normalized frequency (ωH/Vs) and their values decrease with an increase in the value of ωH/Vs. Increase in the damping ratio of soil, increases the value of reliability indices, especially for ωH/Vs values, which are close to π/2. The FORM suffers from few critical shortcomings such as linear assumption of limit state surface at the most probable point of failure and its ability to consider only the statistical uncertainties excluding the effect of epistemic uncertainties. This calls for sampling-based numerical techniques such as Monte-Carlo simulation (MCS) which gives more comprehensive understanding of the problem under consideration in a probabilistic framework. Thus, a computationally efficient surrogate-assisted MCS is carried out to validate the present formulation and provide numerical insights by capturing the system dynamics over the entire design domain. Adoption of the efficient surrogate-assisted approach allowed us to quantify the epistemic uncertainty associated with the system using Gaussian white noise (GWN). Subsequently, its effects on the system reliability index and probabilistic behavior of the critical parameters are presented. The numerical results clearly indicate that it is imperative to take into account the probabilistic deviations of the critical performance parameters for RSRS to ensure adequate safety and serviceability under operational condition while quantifying the reliability of such systems.

Seismic stability analysis for a two-stage slope

This paper adopts the kinematic theorem of limit analysis to assess the seismic stability of a two-stage slope. The seismic effect is taken into account by using the pseudo-static approach. The failure mechanism for the slope is extended to include below-toe failure, toe failure and face failure. Validation of this approach is conducted by comparing the factor of safety with the data in the existing literatures. The stability charts are presented based on the graphical method for reading the factor of safety readily. Parametric study involving the effect of slope geometry, internal friction angle, seismic effect as well as depth coefficient on the stability of a two-stage slope is carried out. The critical failure surfaces with various parameters are plotted. The results obtained reveal the significant influence of slope geometry on the failure mechanism of a two-stage slope under static and seismic condition.

Assessment of the Reinforced Slope Stability Under Earthquake Loading

Failure of the soil slopes by dynamic excitation is one of the most vital geotechnical earthquake hazards which may lead to serious destruction to the bridge abutments, dams, embankment, and structures resting on the slope. Moreover, due to deficient space available for the construction of slopes in urban areas, high and steep slope is constructed by geotextile reinforcement slopes. In this study, numerical modelling is made to study the behaviour of soil slope reinforced by geotextile under earthquake loading using the finite element method available in, optumG2. The value of the factor of safety (FOS) for unreinforced soil slope is calculated using the strength reduction method (SRM) at slope angle (β = 60°) and the height of the slope is 10 m. In the unreinforced slope, a critical failure surface was obtained due to which slope is reinforced with geotextile under static and earthquake loading. Furthermore, a parametric analysis is carried out to evaluate the effect on different lengths of geotextile and different vertical spacing for the stability of reinforced slope with horizontal ground acceleration coefficient (0.1-0.4). From the outcomes of the present study, it is noted that a stable slope can be achieved with an optimized configuration of the reinforcement under seismic loading, also a steeper slope can be achieved using reinforcement compared to that of an unreinforced slope.