Spencer Method Research Articles

A simplified approach to determine the unique direction of sliding in 3D slopes

Based on the Spencer's method, this paper presents a simplified approach to assess the stability of three-dimensional (3D) asymmetrical slopes. The approach allows for satisfaction of the force equilibrium in all three directions and the moment equilibrium about two co-ordinate axes. A unique direction of sliding is involved here to calculate the factor of safety. The direction of sliding (DOS) is always parallel to the plane of symmetry for symmetrical slopes. However, the DOS in asymmetrical slopes could deviate from the symmetrical plane and affect the stability assessments. Through two simple asymmetrical examples, the calculated results demonstrate that the deviation of the sliding from the symmetry could destabilize the slopes and cause failures. Neglecting the DOS in 3D asymmetrical slopes will overestimate their stability. Application of the presented approach into complex asymmetrical problems is straightforward.

Premium-penalty ant colony optimization and its application in slope stability analysis

Ant colony optimization (ACO) is a very suitable path search algorithm, whose typical application is traveling salesman problem. However, as one heuristic algorithm, it has many shortcomings, such as slow convergent speed and low searching efficiency. To overcome these shortcomings, the premium-penalty strategy has been introduced, and the pheromone diversity of the good paths and the ordinary ones is increased to polarize pheromone density of all paths. Thus, premium-penalty ant colony optimization (PPACO) is proposed. And its good performance is verified by the applications to some typical traveling salesman problems. Its two important parameters are discussed too. Because location critical slip surface in slope stability analysis is a path search problem, it can be solved by the ACO very suitably. Therefore, based on PPACO and typical mature limit equilibrium analysis (Spencer method), a new method to analyze the slope stability is proposed. Through two typical examples, one simple slope and one complicated slope, the efficiency and effectiveness of the new algorithm are verified. The results show that, the new algorithm can always find the less safety factor and its critical slip surface in shorter time than many previous algorithms, and the new algorithm can be used in real engineering very well.

Slope Sensitivity Analysis Using Spencer's Method in Comparison with General Limit Equilibrium Method

This paper analyses the sensitivity analysis of a natural-unreinforced slope in Kepong, Kuala Lumpur. In order to conduct a sensitivity analysis, several ranges of data, based on borehole information and figures and also typical range of earth materials geotechnical values are used to set the minimum and maximum value of parameters in Slide 6.0 software. Under Mohr-Coulomb failure criterion, Spencer's and General Limit Equilibrium methods of slices were used in the analysis to determine the influence of varying parameters values towards the change in safety factor. In addition, the percentage differences in safety factors obtained by both methods based on General Limit Equilibrium method are also determined in the analysis. In the analysis, water table location has the highest influence to the change in safety factors of the slope. Besides, the percentage differences in safety factors obtained by both methods are very nominal and showed a good agreement to each other.

Tabu search for critical non-circular slip surface in slope stability analysis

Stability analysis of slopes is of utmost importance because its failure may lead to loss of lives and great economic losses. Locating the critical circular slip surface is the key step of slope stability analysis. Location of the critical slip surface of a general soil slope is difficult because the objective function of the factor of safety is non-convex and multiple minima exist in general. In this paper, a new approach which combines tabu search and limited equilibrium method (Spencer method) was proposed and applied to some classic examples. The results show the proposed search approach is stable and highly correlated with the results of independent analyses. Furthermore, this paper demonstrates the successful application of tabu search to non-circular critical slip surface search problems.

Reliability Analysis of Earth Slopes Considering Spatial Variability

The paper presents a computational procedure for reliability analysis of earth slopes considering spatial variability of soils under the framework of the Limit Equilibrium Method. In the reliability analysis of earth slopes, the effect of spatial variability of soil properties is generally included indirectly by assuming that the probabilistic critical slip surface is the same as that determined without considering spatial variability. In contrast to this indirect approach, in the direct approach, the effect of spatial variability is included in the process of determination of the probabilistic critical surface itself. While the indirect approach requires much less computational effort, the direct approach is definitely more rigorous. In this context this paper attempts to investigate, with the help of numerical examples, how far away are the results obtained from the indirect approach from that obtained from the direct approach. In both the approaches, it is required to use a model of discretization of random fields into finite random variables. A few such models are available in the literature for one-dimensional (1D) as well as two-dimensional (2D) spatial variability. The developed computational scheme is based on the First Order Reliability Method (FORM) coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. The study includes bringing out the computational advantages and disadvantages of the three commonly used discretization models. The sensitivity of the reliability index to the magnitudes of the scales of fluctuation has also been studied.

Probabilistic Stability Analysis of the Bois Brule Levee Considering the Effect of Spatial Variability of Soil Properties Based on a New Discretization Model

The paper presents a computational procedure for reliability analysis of earth slopes for which the probabilistic critical slip surfaces have been determined considering spatial variability of soils. The effect of spatial variability has been taken into account based on a newly developed discretization model which is free from the shortcomings of the discretization models available in the literature and all these shortcomings are demonstrated numerically in this paper. The developed algorithm is based on the First Order Reliability Method coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. A case study of the Bois Brule Levee has been reanalyzed using the developed technique and several observations have been discussed. The result obtained in this study clearly shows that the proposed model is considerably more conservative than other available models.

Mechanism and Remediation of a Seismically Induced Landslide with a Potential for Deep Seated Sliding

This paper presents a study of the failure mechanism and control measures for a seismically induced landslide along the Wanzhou-Liangping expressway in China. It is shown that the dominant cause of the slope stability problem is the original inappropriate geometry of the slope. Slope stability analyses were performed using Spencer's method incorporated with the pseudostatic method, which showed that the rock slope was unstable after the Wenchuan earthquake, and the stability of the studied area was governed mainly by the significant potential of deep sliding. Measures to stabilize unstable zones include resloping and unloading the slope, installing rock anchors and a retaining wall, improving drainage, and vegetating bare areas. The results indicate that the measures were effective in controlling the landslide.

Limit Equilibrium and Limit Analysis: Comparison of Benchmark Slope Stability Problems

AbstractConventional stability analyses use various limit equilibrium (LE) methods to determine the minimum, critical factor of safety and its associated failure mechanism. These methods generally assume that collapse will follow predefined geometric constraints that are dependent on certain input criteria—an effective approach for simple geotechnical problems, but challenging in consideration of complex problems. An effective solution lies in the use of upper bound limit analysis (LA) in conjunction with a discretization procedure known as discontinuity layout optimization (DLO). Use of DLO-LA can be an effective tool for establishing a critical failure mechanism and its stability without the constraints or assumptions required in LE analyses. This study compares the use of LE (Spencer method with dynamic programming optimization, Morgenstern-Price, Spencer) and LA for several examples that focus on complex geotechnical scenarios to illustrate agreement and differences between the analyses, as well as si...

System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms

This paper presents a new approach to identify the fully specified representative slip surfaces (RSSs) of layered soil slopes and to compute their system probability of failure, Pf,s. Spencer's method is used to compute the factors of safety of trial slip surfaces, and the first order reliability method (FORM) is employed to efficiently evaluate their reliability. A custom-designed genetic algorithm (GA) is developed to search all the RSSs in only one GA optimization. Taking advantage of the system aspects of the problem, such RSSs are then employed to estimate the reliability of the slope system, and a proposed linearization approach—based on first or second order (FORM or SORM) reliability information about the identified RSSs—is used to efficiently estimate Pf,s. Three typical benchmark-slopes with layered soils are adopted to demonstrate the efficiency, accuracy and robustness of the suggested procedure, and advantages of the proposed method with respect to alternative methods are discussed. Results show that the proposed approach provides reliability estimates that improve previously published results, emphasizing the importance of finding good RSSs—and, especially, good (probabilistic) critical slip surfaces that might be circular or non-circular—to obtain good estimations of the reliability of soil slope systems.

Slope stability analysis based on immunised evolutionary programming

The key task in analyzing slope stability is determining its critical slip surface. To solve this problem, a new critical slip surface search method based on an immunised evolutionary programming is proposed. From application of the improved adaptive mutation operation and improved selection operation based on thickness adjustment of artificial immune system into evolutionary programming, a new immunised evolutionary programming for continuous optimisation is proposed. Then, combined with typical mature limit equilibrium analysis (the Spencer method and the Morgenstern–Price method), the immunised evolutionary programming is used to analyze the stability of the slope. Finally, the proposed method is verified through four typical numerical examples and is compared with the evolutionary programming and previous studies in the literature. The results indicate that the applicability, validity and effectiveness of the immunised evolutionary programming are all improved over previous studies and over the evolutionary programming.

Seismic stability and displacement analyses of earth slopes using non-circular slip surface

The seismic stability and deformability of earth slopes are conventionally evaluated by simple, practical methods. Because a multimodal function optimization problem makes it mathematically difficult to search large critical slip surface of earth slopes with complex strata, stability analysis is one of the classical problems of geotechnical engineering. One option is to evaluate the seismic deformability of earth slopes using permanent seismic displacements via Newmark׳s sliding block analysis in the current seismic design. The advantage of this method is that it is useful in practice and is less time consuming in terms of calculations. However, the calculations require that the critical slip is assumed either linear or circular. This paper proposes two methods for computing safety factors and permanent seismic displacements of earth slopes using an efficient non-circular slip surface search algorithm based on the force equilibrium given by the Spencer method. The validity of these proposed methods is verified by applying them to models with a known safety factor or theoretically calculated permanent seismic displacement and the results obtained compared. Comparative analyses are also conducted in order to demonstrate their efficacy in terms of computation precision and convergence performance. Further, they are utilized to calculate the permanent seismic displacement of a practical earth slope model subjected to seismic motions in both the horizontal and vertical directions. The results obtained indicate that they can calculate the safety factor of earth slopes using a smaller number of simulations than conventional methods and that they can also be applied to calculate the permanent seismic displacement of earth slopes. The results also indicate that the permanent seismic displacement calculated is an important index that can be used to quantitatively evaluate the seismic performance of earth slopes.

Slope Stability Analysis Using Radial Slices: A Mathematical Model

In this paper, a mathematical model has been formulated for calculating the factor of safety of a slope. Corresponding computer code has also been developed. Limit equilibrium method (moment equilibrium) has been adopted for calculating the net resulting driving and resisting forces. The probable slip circle region has been divided into radial slices for the simulation process. In this approach, the inter-slice shear forces are zero. Thus, the calculation process becomes simpler as compared to that with vertical slices. The slope stability analyses were done. Validation of the present program was done with existing limit equilibrium based methods. Various models were prepared and analysed with varying geometry and soil strength parameters. These models were also analysed with other limit equilibrium methods like Bishop, Janbu and Spencer method. The results were found to be in agreement with the results of other limit equilibrium methods for the same dump soil properties and slope parameters.

Application of dynamic programming to locate the critical failure surface in a rainfall induced slope failure problem

In this paper, a dynamic programming method is employed in conjunction with limit-equilibrium techniques to determine the location of the non-circular critical slip surface in the analysis of a slope failure due to a rainfall event. The Spencer method of slope stability analysis was incorporated into dynamic programming to predict the time of a slope failure and shape of the failure surface. A one-Dimensional (1D) sliding block model was used to analyze the motion of the failure mass. Furthermore, during the movement of the sliding mass, the stability of the model slope was analyzed by updating the shape of the model slope according to the new position of the sliding mass. The suction head in the soil pore provides soil shear strength to maintain the stability of the slope. The positive pore water pressure reduces the soil shear strength so that the slip surface beneath the water table will be more unstable. Numerous studies (using non-cohesive soil) have been conducted to locate the critical slip surface in slope stability problems without considering the increase in shear strength due to suction and apparent cohesion. The studies do not clearly illustrate the reasonable boundary conditions to avoid the slip surface alignment totally outside the actual slope. This paper clearly discusses the boundary conditions for such a slope stability problem reasonably. The step-by-step calculation procedure is presented through the flow diagram. Three cases of slope failure obtained from a laboratory flume experiment have been analyzed using the proposed method. The numerical simulation results and the results obtained from experiments are comparable.

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.

Probabilistic Critical Slip Surface for Earth Slopes Based on the First Order Reliability Method

The paper presents a computational procedure for reliability analysis of earth slopes in which the probabilistic critical slip surface is located using a mathematical programming formulation similar to that used to search for the deterministic critical slip surface in a conventional slope stability analysis. The procedure is based on the First Order Reliability Method (FORM) in conjunction with the Spencer method for evaluation of factor of safety of general slip surfaces and the Sequential Quadratic Programming (SQP) technique of optimization. When applied to two benchmark illustrative examples, the procedure yields the probabilistic critical slip surfaces which are reasonable and the values of the associated minimum reliability indices are close to or lower than those reported by different investigators using different methodologies, and compare well with those from the direct Monte-Carlo simulation method (MCS). Further, the developed procedure has been made use of to investigate another important aspect of reliability analysis, namely, how the results of reliability analyses vary with the probability distributions assumed for the basic random variables. Three most commonly assumed distributions, namely, normal, lognormal and truncated normal distributions have been considered. The results indicate that the lognormal assumption used solely to ensure non-negativity of the basic variables might lead to non-conservative prediction of probability of failure. Whether the effect of such distributional assumption is consistent when the coefficients of variation (COVs) of the basic random variables vary within their respective ranges has also been investigated.

FORM, SORM, and spatial modeling in geotechnical engineering

An intuitive ellipsoidal perspective is described together with three spreadsheet-automated constrained optimizational FORM procedures and a SORM approach. The three FORM procedures are then compared in the context of geotechnical examples of a confined soil element, a rock slope, and an embankment on soft ground with spatially autocorrelated undrained shear strength in the soft clay foundation, the performance function of which is based on a reformulated Spencer method with search for reliability-based critical noncircular slip surface. Two methods of modeling spatial autocorrelations are presented, and the merits and limitations of the three constrained optimizational FORM procedures are studied. The complementary roles and interconnections among the three constrained optimizational FORM procedures and SORM approach are emphasized. Comparisons are also made with Monte Carlo simulations.

Analysis on failure of slope and landslide dam

A three-dimensional (3D) seepage-flow numerical simulation model was developed for seepage analysis of a landslide dam. A 3D seepage-flow numerical simulation model coupled with a two-dimensional (2D) surface flow and erosion/deposition model was developed for seepage analysis of a slope due to a rainfall event. The conventional water-phase (one-phase) seepage-flow model assumed only water-phase flow in seepage analysis, which was inadequate for unsaturated soil domains. A water-air two-phase seepage-flow model that considers both the water and air phases in the seepage-flow process was also used for the seepage analysis. The pore-water pressure and moisture-content data obtained from the seepage-flow model were used to analyze the stability. Janbu’s simplified method and the extended Spencer method were used for the stability analysis. The numerical simulation results almost compared well with laboratory experimental measurements.

An artificial bee colony algorithm for locating the critical slip surface in slope stability analysis

Determination of the critical slip surface with the minimum factor of safety of a slope is a difficult constrained global optimization problem. In this article, an artificial bee colony algorithm with a multi-slice adjustment method is proposed for locating the critical slip surfaces of soil slopes, and the Spencer method is employed to calculate the factor of safety. Six benchmark examples are presented to illustrate the reliability and efficiency of the proposed technique, and it is also compared with some well-known or recent algorithms for the problem. The results show that the new algorithm is promising in terms of accuracy and efficiency.

Stability assessment of earth slope using modified particle swarm optimization

This paper has proposed an effective method to determine the minimum factor of safety (FS) and associated critical failure surface in slope stability analysis. The search for the minimum FS based on limit equilibrium methods is a complex optimization problem as the objective function is non-smooth and non-convex. Recently, particle swarm optimization (PSO) as a meta-heuristic optimization algorithm has been developed with success in treating various types of problems. In the current study, a new approach of PSO is proposed to calculate the safety factor of earth slopes. The safety factors of the general slip surfaces are calculated using Spencer method of slices, and each new slip surface is randomly generated by straight line technique. The reliability and efficiency of the proposed algorithm are examined by considering a number of published cases. The results indicate that the new method can predict a more critical failure mechanism with a lower FS and can outperform the other methods in the literature as well as standard PSO. Finally, the proposed method will be validated by considering an existing slope failure in Ulu Klang, Malaysia.

Improved slope safety analysis by new Druker-Prager type criterion

Based on Mohr-Coulomb (M-C) criterion, the parameters of Druker-Prager (D-P) criterion for geomaterial were determined under non-associated flow rule, and thus a new D-P type criterion was presented. Two assumptions were employed during the derivation: 1) principal strains by M-C model and D-P model are equal, and 2) the material is under plane strain condition. Based on the analysis of the surface on π plane, it is found that the proposed D-P type criterion is better than the D-P criterion with M-C circumscribed circle or M-C inscribed circle, and is applicable for stress Lode angle less than zero. By comparing the predicted results with the test data of sand under plane strain condition and other D-P criteria, the proposed criterion is verified and agrees well with the test data, which is further proved to be better than other D-P type criteria in certain range of Lode angle. The criterion was compiled into a finite difference package FLAC3D by user-subroutine, and was used to analyze the stability of a slope by strength reduction method. The predicted slope safety factor from the proposed criterion agrees well with that by Spencer method, and it is more accurate than that from classic D-P criteria.