Limit Equilibrium Solution Research Articles

Uncoupled analysis of piles stabilized transmission tower slopes

It is a common engineering practice to adopt piles to improve the stability of potentially unstable slopes. In this paper, parametric analysis of a homogeneous transmission tower slope stabilized with a row of small diameter steel piles is conducted using an uncoupled numerical method in which the pile response and slope stability are considered separately. The transmission tower slope stability is analyzed using the limit equilibrium solutions, and the stabilizing pile-soil system is modeled as a simple beam on subgrade reaction foundation. Attempts have been made to study the effect of soil property, slope geometry, pile property, pile spacing and pile position on the safety factor of the transmission tower slope stabilized with small diameter steel piles. The study shows that they have a combined effect on the performance of small diameter steel piles to stabilize transmission tower slopes.

Rigorous Limit-Equilibrium Solution for Three-Dimensional Slope Stability with an Asymmetric Slip Surface

Rigorous Limit-Equilibrium Solution for Three-Dimensional Slope Stability with an Asymmetric Slip Surface

A practical approach for predicting landslide retrogression and run-out distances in sensitive clays

In the present study a novel approach of practical interest is proposed to perform a predictive analysis of the landslides in sensitive clays. Specifically, this approach allows a prediction of the landslide triggering along with the consequent distances of retrogression and run-out. To this end, a simple limit equilibrium solution is proposed to deal with the first two steps (prediction of the landslide triggering and the subsequent retrogression distance). A calculation code based on the material point method (MPM) is instead employed for evaluating the run-out distance of the unstable soil mass that was defined in the previous step. Operative values of the undrained shear strength are also suggested for each of the aforementioned steps. The proposed method is very attractive from a practical viewpoint because it is simple to use, not time-consuming and requires few geotechnical parameters of easy experimental determination as input data. The method is applied successfully to several well-documented case studies available in the literature, demonstrating also its reliability for engineering applications. Three of these landslides are analysed in more detail to illustrate the application of the present method to real cases.

On the performance-based seismic design of yielding retaining structures

This work summarises recent research on the seismic behaviour of displacing or yielding retaining structures, i.e., structures that can undergo permanent displacements during strong earthquakes without failing. For these systems, energy dissipation on shaking, leading to reduced inertia forces, can be achieved by allowing the activation of ductile plastic mechanisms. These must be correctly identified to guarantee the desired strength hierarchy, and depend on the specific retaining structure under examination. It is shown that the critical acceleration, or the smallest value of acceleration corresponding to the activation of the critical plastic mechanism, is the key ingredient for the performance-based design of yielding retaining structures. In fact, the critical acceleration controls both the maximum internal forces in the structural elements and the magnitude and trend of post-seismic permanent displacements and rotations, required for quantitative serviceability and post-earthquake operability assessment of infrastructures. Based on a clear understanding of the physical mechanisms governing the dynamic behaviour of these systems, pseudostatic limit equilibrium solutions and simplified dynamic methods can be developed for their seismic design. Theoretical predictions are validated against data from reduced scale centrifuge models and results of pseudo-static and fully dynamic numerical analyses. Finally, all the results presented in the paper, including experimental, numerical and theoretical findings, are used to provide suggestions for the performance-based design of retaining structures.

Solving the general 3-D safety factor by combining Sarma's idea with the assumption of normal stress distribution over the slip surface.

This study proposes a method for determining 3-D limit equilibrium solutions. The method, inspired by Sarma, introduces the horizontal seismic coefficient as a slope failure parameter and implements a modification of the normal stress over the slip surface. Four equilibrium equations are used to solve the problem without compromising the accuracy of the calculations: three force equilibrium equations in the x, y, and z directions and a moment equilibrium equation in the vertical (z) direction. The reliable factor of safety can be determined by calculating the minimum value of the horizontal seismic coefficient. Furthermore, we analyzed several typical examples of symmetric and asymmetric slopes, finding good consistency with the existing literature. This consistency indicates the reliability of the factor of safety we obtained. The proposed method is favored due to its straightforward principle, convenient operation, fast convergence, and ease of programming.

Centrifuge and theoretical study of the seismic response of anchored steel sheet pile walls in dry sand

Displacement-based approaches have been proven to be effective for the seismic design of gravity retaining structures. However, application of these methods to embedded flexible walls, such as anchored steel sheet pile (ASSP) walls, is still an open issue, as the factors affecting the accumulation of permanent displacements of these systems are not fully understood. This paper presents the results of four dynamic centrifuge tests on small-scale models of ASSP walls in uniform dry sand. The experimental results showed that ASSP walls can experience different failure mechanisms depending on their geometry and on the magnitude of accumulated displacements. During the earthquake, the system exhibits an overall increase of its seismic capacity, caused by: (a) progressive mobilisation of the soil passive strength; (b) sand densification; (c) rotation of the walls; and (d) reduction of the retained height. Accounting for these effects in pseudo-static limit equilibrium solutions improved the predictions of both the critical failure mechanism and the maximum internal forces in the structural members. Moreover, it allowed development of an efficient analytical tool for the prediction of the seismic permanent displacements of these systems, based on Newmark's sliding block approach.

Numerical and analytical investigation on inclined plate anchors in sand

The paper investigates the ultimate capacity of inclined plate anchors in sand through a series of finite element (FE) analyses. The FE simulations adopted a sophisticated bounding surface plasticity model that accounts for stress and density dependent behaviour, thus portraying a realistic estimate of the shear stresses and shear strains within the failure mechanism. The simulations considered a strip anchor with different plate widths, plate inclination angles and shallow embedment ratios in dense sand. The FE results show good agreement with previously reported experimental data. A limit equilibrium solution is subsequently proposed to predict capacity of inclined strip anchors in sand, based on mathematical expressions of the FE simulated normal and shear stresses along mobilised failure planes. The limit equilibrium solution is shown to produce satisfactory agreement with results from a large database of inclined plate anchors.

Experimental investigation on pipe-soil interaction due to ground subsidence via high-resolution fiber optic sensing

Understanding the pipe-soil interaction is critical for the design and safety assessment of buried pipelines subjected to large differential ground motion. Laboratory model tests were conducted to study the mechanical response of polyvinyl chloride pipes to ground subsidence. The optical frequency domain reflectometry technique was employed to perform distributed strain sensing of the pipe-soil system. An analytical method was proposed to estimate greenfield settlements in terms of fiber optic strain measurements. The pipe-soil interaction mechanism due to ground subsidence was investigated concerning four aspects, i.e., displacement profile of the pipe-soil system, pipe bending, soil arching, and interfacial shearing. Accounting for three-dimensional soil arching effects, a limit equilibrium solution for the earth pressure imposed on the pipe crown was given. This solution was developed from the two-dimensional Terzaghi soil arching theory and was validated based on the measured earth pressures. The feasibility of distributed fiber optic strain sensing technique in monitoring pipe-soil deformation was discussed through the comparison with the data of displacement transducers. In addition, the change in relative pipe-soil stiffness with increasing ground movements was studied based on the bending behavior of the pipe. A simplified method was given for quantifying the axial pipe-soil interaction based on longitudinal strains. This paper presents new inspiration on the intricacies of pipe-soil interaction using high-resolution distributed fiber optic sensing, which can be applied to structural health monitoring and optimum design of buried pipelines.

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

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

Ultimate pullout capacity of single vertical plate anchors in sand

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

Limit equilibrium solutions to anti-overturning bearing capacity of suction caissons in uniform and linearly increasing strength clays

Suction caissons supporting offshore wind turbines are exposed to great horizontal loading above the soil surface, which may lead to overturning failure of the caisson. This paper presents a modified three-dimensional failure mechanism to analyze the anti-overturning bearing capacity of suction caissons. The modified failure mechanism is composed of a meniscus-conical wedge with a meniscus shape at the soil surface and a scooped shape. The analytical solution to the anti-overturning bearing capacity of the suction caisson is deduced in terms of the limit equilibrium method, followed by a parametric study of the wedge depth ratio (c) to optimize the critical failure mechanism that satisfies both the force and moment equilibriums. Thus, the methodologies are relatively easy to implement in traditional spreadsheets, and the analyses tend to perform very quickly. The effects of gap formation at the rear side of the caisson, loading eccentricity, and adhesion factor at the caisson–soil interface on the anti-overturning bearing capacity are also investigated. Compared with the finite element limit analysis results, experimental data, and existing theoretical solutions, it is proved that the presented limit equilibrium analysis can satisfactorily predict the anti-overturning bearing capacity of suction caissons with low aspect ratios for offshore wind turbines in uniform and linearly increasing strength clays.

The Hydromechanical Interplay in the Simplified Three-Dimensional Limit Equilibrium Analyses of Unsaturated Slope Stability

This paper presents a three-dimensional slope stability limit equilibrium solution for translational planar failure modes. The proposed solution uses Bishop’s average skeleton stress combined with the Mohr–Coulomb failure criterion to describe soil strength evolution under unsaturated conditions while its formulation ensures a natural and smooth transition from the unsaturated to the saturated regime and vice versa. The proposed analytical solution is evaluated by comparing its predictions with the results of the Ruedlingen slope failure experiment. The comparison suggests that, despite its relative simplicity, the analytical solution can capture the experimentally observed behaviour well and highlights the importance of considering lateral resistance together with a realistic interplay between mechanical parameters (cohesion) and hydraulic (pore water pressure) conditions.

Three-Dimensional Limit Equilibrium Solution of Minimum Support Pressure of Shield Tunnel Face in Sandy Cobble Stratum

The ground subsidence accident caused by the instability of shield tunnel face has been a major problem that puzzles urban subway construction in weak rock. Maintaining the stability of tunnel face is the key to ensure the safety of shield construction and reduce the environmental impact. In view of the shield with a diameter of 8600 mm in Chengdu metro line 18 through the sandy cobble stratum, it is more difficult to maintain the stability of the shield tunnel face than other strata. The characteristics of the sandy pebble stratum were taken into account firstly in this paper. The analytical equation was obtained by the limit equilibrium method. The optimization of the trapezoidal bottom has been improved based on the existing 3D trapezoidal wedge model. And the optimal solution of the ultimate support pressure was obtained. The influencing factors of support pressure are also analysed. When the tunnel diameter is fixed, the deeper the ground covering depth is, the greater the required limit support pressure becomes. But the ground covering depth has almost no effect on the ultimate support pressure even if the tunnel is deep. The limit support pressure is inversely proportional to the internal friction angle of the soil and is proportional to the diameter of the tunnel. Finally, theoretical limit support pressure can be also verified by the numerical simulation results and the field monitoring results. It provides a reference for the subsequent setting value of the soil warehouse pressure.

Towards a simple and reliable method for calculating uplift capacity of plate anchors in sand

This paper investigates the uplift capacity of horizontal plate anchors in sand through finite element analyses and centrifuge experiments. The finite element simulations adopt a sophisticated bounding surface plasticity model that accounts for stress- and density-dependent behaviour, as well as loading- and fabric-related anisotropic effects in sands. Failure mechanisms at peak anchor capacity show that failure occurs progressively, with a marked decrease in mobilised friction angle within the shear bands close to the anchor edge. Numerical simulations of a large set of centrifuge experiments on rectangular, strip, and circular plates at different relative densities and stress levels are in good agreement for dense conditions, but perform more poorly for loose conditions due mainly to the open cone yield surface in the bounding surface model. Equivalent comparisons with current limit equilibrium methods highlight the challenges in direct application of element-level strength equations. Finally, the paper proposes a modified limit equilibrium solution based on a “rigid-block” failure mechanism extending to the soil surface, but with anchor factors that encompass the results from the finite element simulations. The modified solution provides a higher level of agreement with results from a large database of plate and pipeline test data than existing limit equilibrium methods.

Limit equilibrium solution for the rock slope stability under the coupling effect of the shear dilatancy and strain softening

In addition to obeying the generalized Hoek-Brown (GHB) strength criterion, the shear behavior of the rock mass sometimes has the effect of the shear dilatancy and characteristics of the strain softening. However, the traditional limit equilibrium (LE) methods can only solve the slope factor of safety (FOS), and it is difficult to calculate the displacement of sliding body (DOSB) of slope. Therefore, it is rarely reported that the LE analysis on rock slope stability under the coupling effect of the shear dilatancy and strain softening. Here, referred to the effect of the shear dilatancy and characteristic of the strain softening for rock mass, the calculation model for slope DOSB is constructed. Then, based on the assumptions of the stresses on the slip surface, the LE solution for rock slope stability is established according to the mechanical equilibrium conditions satisfied by the sliding body. Meanwhile, the calculation formula of slope DOSB is derived by introducing the energy balance equation of sliding body for the slope with the small sliding displacement. Further, the iterative loop calculation strategy is given to solve the slope FOS and slope DOSB for achieving the purpose of quickly obtaining the implicit solution. By comparison and analysis on some slope examples, the feasibility of the present method is verified. Compared to the traditional LE methods, the present method can not only analyze the stability of rock slope under the nonlinear strength criterion, but also study the influence of the shear dilatancy and strain softening of rock mass on the rock slope stability.

Study on the Ultimate Supporting Force of Shield Excavation Face Based on Anisotropic Strength Theory

The determination of the ultimate supporting force of the shield excavation face is an important problem to be solved in shield construction. Considering that the tunnel burial depth ratio has a significant effect on the instability mode of the excavation face, the classic “wedge-prism” limit equilibrium model is improved. Based on the rotation effect of principal stress axis, the Casagrande anisotropic strength equation is introduced into the modified limit equilibrium model of “wedge-prism”, and then the limit equilibrium solution of the ultimate supporting force of shield excavation face in anisotropic soil is deduced. Finally, the influence of each calculation parameter on the ultimate supporting force is analyzed by examples. The research results show that the results of the modified “wedge-prism” calculation model proposed in this paper are slightly larger than those of the centrifugal test. If the influence of the instability mode of excavation face and the anisotropy of soil strength on ultimate supporting force of the shield excavation face is not taken into account, the calculation result will be unsafe. The limit supporting force of shield tunnel excavation surface has a simple linear relationship with the anisotropy ratio. When the anisotropy ratio is greater than 1, the ultimate supporting force of shield excavation face decreases first and then tends to be stable with an increase in the buried depth ratio. When the anisotropy ratio is less than 1, the law is reversed. The more obvious the anisotropy of soil strength, the greater the rate of change of ultimate supporting force. The limit supporting force of the shield excavation face decreases linearly with the exertion of loosening earth pressure, linearly decreases with the increase in soil cohesion, and decreases nonlinearly with the increase in the angle of internal friction in soil. The relevant conclusions will provide theoretical guidance for controlling the reasonable chamber pressure of shield tunneling, and ensure the safety of construction.

Numerical limit analysis of the ultimate load bearing capacity of a pier foundation under complex load

A pier foundation has plenty of advantages as the foundation form for large onshore wind turbines in the terrain of the Gobi desert and collapsible loess areas. The ultimate load bearing capacity design, as an important part in the design phase of this foundation form, is not sufficiently in terms of design theory and the checking method, especially for application in collapsible loess areas. In this paper, numerical limit analysis has been employed to analyze the load bearing capacity for the scheme of selection of a pier foundation of a 100 MW wind farm in certain collapsible loess geology in Ningxia. The results were then compared with the empirical formulas, the limit equilibrium solutions and the finite element solutions to verify the accuracy of the results. It has been demonstrated that under the ultimate engineering load, the pier foundation can meet the stability requirement. In the ultimate state, whether the surrounding soil of the foundation falls in plasticity is associated with the ratio of the depth to the diameter of the foundation. The numerical limit analysis method can automatically determine the failure surface.

Forecasting vulnerability of deep extraction level excavations to draw-induced cave loads

Block and panel caving methods are increasingly used for mining of large and strong orebodies at large depths below the ground surface (>1 km). This paper focuses on the production phase of caving when the extraction level is fully developed and subjected to the weight of overlying caved material. A limit equilibrium solution for estimation of cave loads is critically reviewed and combined with existing empirical tools in order to forecast extraction level performance under cave loading. The analysis results, presented in nomogram form, suggest that the risk of severe squeezing of extraction level tunnels does not increase with cave depths below 500–1000 m due to the beneficial effects of arching. Under normal extraction ratios (ER = ∼50%), severe squeezing experienced from cave loads in deep mines appears to be unlikely in all but the weakest rock masses (σcm < 5 MPa, where σcm is the rock mass global strength), even with poor draw control or low drawpoint availability. The likelihood of severe squeezing is greater when large draw areas (HR = 50–75 m, where HR is the hydraulic radius) are combined with low drawpoint availability or poor draw control and locally higher extraction ratios (ER = ∼75%), even in better quality rock (σcm = 5–10 MPa). The analysis results are back-analyzed with the extraction level performance before and after a doubling production of rate and draw area at the deep ore zone (DOZ) mine in Indonesia. The extraction level conditions predicted by the nomograms correlate well with the experience at the DOZ, in which extraction level damage increases significantly over the production rampup, accompanied by a significant drop in drawpoint availability.

Effect of combined translation and torsion on undrained uplift capacity of plate anchors: Plastic limit analysis (PLA) solution

Uplift capacity of plate anchors have been the focus of numerous studies, since anchor plates are designed for pull out in normal operating conditions. However, the response of plate anchors under six-degrees-of-freedom loading caused during extreme loading conditions is poorly understood. The purpose of this study is to propose a simple yet sufficiently accurate analytical solution to investigate the behavior of plate anchor under combined in-plane translation and torsion and to evaluate its effect on the plate uplift bearing capacity. To this end, a modified plastic limit analysis (PLA) approach is introduced and compared with limit equilibrium (LE) and simplified upper bound baseline solutions. The proposed method is verified with three dimensional finite element (3D-FE). The variables considered in this study include plate aspect ratio, plate thickness, as well as load direction and eccentricity. Results of analytical solutions indicate the insensitivity of the “shape” of the shear-torsion yield envelope to plate thickness. This finding facilitates the use of simplified yet reasonable yield envelope for infinitely thin plate obtained from simplified PLA approach for other plate thicknesses. The “size” of the failure envelope (controlled by pure torsional and translational capacity) could be predicted fairly accurately by PLA and LE methods. Combination of these analytical methods offers a simple yet reasonably accurate solution to describe shear torsion response of anchor plate. The obtained shear-torsion yield envelope is then fitted in the generalized six-degrees-of-freedom yield surface which describes the reducing effect of moment, torsion, and planar forces on the uplift capacity of plate.

Numerical modelling of slope–vegetation–atmosphere interaction: an overview

The behaviour of natural and artificial slopes is controlled by their thermo-hydro-mechanical conditions and by soil–vegetation–atmosphere interaction. Porewater pressure changes within a slope related to variable meteorological settings have been shown to be able to induce soil erosion, shrinkage–swelling and cracking, thus leading to an overall decrease of the available soil strength with depth and, ultimately, to a progressive slope collapse. In terms of numerical modelling, the stability analysis of partially saturated slopes is a complex problem and a wide range of approaches from simple limit equilibrium solutions to advanced numerical analyses have been proposed in the literature. The more advanced approaches, although more rigorous, require input data such as the soil water retention curve and the hydraulic conductivity function, which are difficult to obtain in some cases. The quantification of the effects of future climate scenarios represents an additional challenge in forecasting slope–atmosphere interaction processes. This paper presents a review of real and ideal case histories regarding the numerical analysis of natural and artificial slopes subjected to different types of climatic perturbations. The limits and benefits of the different numerical approaches adopted are discussed and some general modelling recommendations are addressed.