Limit Analysis Method Research Articles

Upper bound analysis of longitudinally inclined EPB shield tunnel face stability in dense sand strata

Earth pressure balance (EPB) shield machine frequently drives along a route with a longitudinal inclination angle (δ) during tunnel construction. The tunnel face stability is a crucial issue for controlling the safety of tunneling. In this paper, the finite difference method (FDM) is used to investigate the face stability of longitudinally inclined shield tunnel in dense sand strata. Based on the numerical results of soil failure pattern in front of tunnel face, a novel two-dimensional (2D) failure mechanism is conceived following the upper bound theorem. It is assumed that the failure zone consists of an upper loosen zone, an arc-shaped shear band, and a triangular rigid body, which can calculate the active limit face support pressure pu. A scale factor for pu is introduced to quantify the three-dimensional (3D) behavior from the 2D calculation, with respect to different analysis approaches, e.g., numerical simulation, limit analysis, and limit equilibrium method. By incorporating the measure of scale factor, the calculation accuracy of the proposed 2D failure mechanism can be improved to provide reasonable prediction for longitudinally inclined tunnel.

Research on Upper Bound Limit Stability Analysis of Multi-Step High Steep Slope

In order to explore the stability of multi-step high steep slope, this paper considers layer distribution of soil and builds a slope failure mechanism which meets requirements of velocity separation based on plastic mechanics principles, then derives the calculation method of external force power and internal energy dissipation power in failure area using numerical analysis theory, and proposes a loop algorithm for slope stability analysis combining upper bound limit analysis theorem and strength reduction principle. At the same time, in order to enhance the applicability of the upper bound limit analysis method, this paper develops a slope stability calculation system of upper bound limit analysis combined with computer programming technology and evaluates the accuracy of the calculation system through slope engineering practice in open-pit mine. The results show that stability factor by calculation system of upper bound limit analysis is an upper limit solution compared with limit equilibrium method, but the error rate of two methods is less than 5%, which indicates that the slope stability calculation system established in this paper has high accuracy in stability analysis of multi-step high steep slope in the practice of open-pit mine. On the other hand, the most dangerous sliding surface searched by upper bound limit analysis can fully solve velocity problems of multi-step high steep slope failure, so it has stronger reference values in engineering practice.

A general kinematic approach to the seismic stability assessment of slopes

Most of the current studies concerning slope stability assume a toe failure mechanism, which fails to capture the full complexity of slope collapse. In this study, a more comprehensive approach is developed for analyzing seismic slope stability by defining an instability behavior coefficient that considers different failure patterns, including toe, face, and base failure. The proposed coefficient is integrated with a kinematic method of limit analysis to establish a three-dimensional (3D) collapse mechanism that can describe all possible instability behaviors. The impact of earthquake forces on instability behaviors and slope stability is assessed in a modified pseudodynamic manner, which is capable of accounting for the spatiotemporal feature of seismic acceleration. A horizontal slice method is executed for calculating earthquake-induced work rates, as well as a double integral technique to compute dissipation rates due to soil resistance along the sliding surface and work rates generated by soil weight. By applying the principle of energy conservation, the lower-bound solutions of stability numbers are derived, and a further critical solution can be obtained by the combination of the Marine Predators Algorithm and the Nelder-Mead Simplex Algorithm. The proposed method is confirmed to be accurate and effective through comparisons with previous solutions. In addition, a parametric study reveals the effect of the dynamic characteristics of seismic waves on slope stability and instability behaviors. The general approach developed in this study offers a more comprehensive and accurate way to assess seismic slope stability.

A Physically Based Model for the Sequential Evolution Analysis of Rainfall‐Induced Shallow Landslides in a Catchment

AbstractPredicting the occurrence time, volume, distribution and run‐out length of shallow landslides is critical for assessing the volume of debris flow in a catchment. Recent studies suggest that the landslides hydro‐mechanical triggering (LHT) model based on threshold‐based mechanical interactions can adequately predict rainfall‐induced landslides. However, this model assumes that the soil‐rock interface is the only sliding surface and cannot dynamically determine a sliding surface above the soil‐rock interface. Therefore, by calculating the wetting front depth in the shallow soil layer and using the limit analysis method, the most likely sliding surface can be dynamically calculated to improve the model. Through linking the Soil and Water Assessment Tool (SWAT, USDA) model and the depth‐resolved LHT model (D‐LHT), a framework for predicting shallow landslides in a catchment was proposed in this work. This framework considers the effects of antecedent rainfall, the mechanical reinforcement of roots, and the spatial distribution of soil properties on slopes. The D‐LHT model was applied to the Jiangjia gully in Yunnan, China, to adequately predicted the occurrence time, scale and spatial distribution of shallow landslides. In addition, the present and antecedent rainfall effects on the shallow landslide volume were analyzed. The results showed that the average soil moisture content threshold was determined by antecedent and same‐day rainfall, which affect the timing and volume of the shallow landslides. This study provides a new method for predicting shallow landslides. The stability and simplicity of the model make it suitable for early warning systems.

Upper bound solution of passive instability on the face of longitudinal inclined shallow buried shield tunnel based on rotation–translation mechanism

To better evaluate the passive failure mode of shallow shield tunnel faces in sandy soil strata, tunnel faces passive failure mode was investigated considering the longitudinal inclination in this study. A three-dimensional rotation–translation mechanism that simultaneously considers the longitudinal inclination of the tunnel and partial failure of the tunnel face based on the upper bound limit analysis method was presented. The mechanism comprised two rigid translation blocks and one rigid rotation block. Subsequently, the reliability of the proposed mechanism was verified using the horizontal rigid translation mechanism and numerical simulation considering the longitudinal inclination. Finally, the effects of tunnel longitudinal inclination (δ), cover depth ratio (C/D), soil internal friction angle (ϕ), and the possible presence of uniform surface surcharge (σS) on the limit support pressure of the tunnel face and the partial failure range was assessed. The results indicated that the partial failure range of the tunnel face gradually increased with δ When C/D<1. Conversely, global failure of the tunnel face occurred at different longitudinal inclinations when C/D≥1. The rotation angle (θ) of the rotational failure zone was sensitive to the longitudinal inclination (δ) and decreased with δ at the same coverage depth ratio.

Effect of single and dual voids on lateral capacity of large-diameter monopiles in marine clay

This study numerically examines the effect of single and dual voids on the lateral bearing capacity of nearshore large-diameter monopiles in marine clay by the finite element limit analysis (FELA) method. The ability of the numerical model to accurately predict the pile-void system behavior is evaluated by performing degenerated verification analyses on existing studies. Based on numerous numerical results, several design charts for capacity reduction coefficients are produced to provide a quick and straightforward assessment of the pile capacity with single and dual voids by multiplying that without voids by coefficients. Particularly, the coupling coefficient is further proposed to quantify the coupling effect between dual voids, and the corresponding design charts are also given. The subsequent parametric studies indicate that the void effect gets aggravated as the void is enlarged and close to piles but would be weakened as the soil around the pile gets strengthened; the void effect is positively and negatively associated with the void depth in the cases of low and great shear strength, respectively. The effect of single and dual voids (inc. coupling effect between dual voids) on the pile capacity and failure mode is further graphically explained by the plastic multiplier contour maps.

Active Earth Pressure against Cantilever Retaining Walls with a Long Relief Shelf in Rotation about the Top

A cantilever retaining wall with a long relief shelf (CRW-LRS) is a particular type of retaining wall, but no studies have yet reported on rotation about the top of this type of retaining wall. This work studied failure mechanisms in semi-infinite non-cohesive soils in rotation about the top using the finite element elastoplastic analysis method (FEEA). By considering the influence of both the relief shelf and rotational stress from the soil, a formula for calculating the active earth pressure (Ea) was developed using the horizontal differential layer limit analysis method. Four different parameters were studied, i.e., unloading plate's geometric conditions, heel length and the internal friction angle of the backfill, in terms of their influence on the Ea distribution, resultant force etc. The lowest Ea was found when the relief shelf' relative position m = 0.4.

Influence of Soil Suction and Seismic Excitation on Active Earth Pressure

Earthquake-induced slope failures are common occurrences in engineering practice. The rigid gravity retaining wall remains an effective technique in maintaining the slope stability. Based on the pseudo-dynamic approach, the seismic response of unsaturated backfills is studied using the kinematic limit analysis method. To formulate the energy balance equation, a horizontal slice method is proposed. The work rates of unsaturated soil gravity, seismic inertia force and damping force that are all characterized with conspicuous nonlinear distribution characteristics are computed using this method. A series of parametric analysis is conducted regarding the influences of soil suction and seismic excitations on the earth pressure estimations. The results show that the suction effect relates not only to the soil types but also the seismic excitations. The earth pressure is overestimated by 15%–32% when soil suction is not considered in the analyses. For hard backfills, the peak active earth pressure increases slightly and moves towards the negative direction.

Analysis of the effects of isolation piles on the basal stability of foundation pits against upheaval based on continuous velocity fields.

The upper bound limit analysis method is one of the main approaches to check the basal stability of foundation pits against upheaval. However, existing studies have often failed to consider the effects of external supporting structures, including isolation piles and others, on basal stability against upheaval. This study derives a formula for the coefficient of basal stability against upheaval under the action of isolation piles by simplifying the pile-soil relationship and systematically examines the impact of isolation pile parameters on basal stability against upheaval using the theory of continuous velocity fields and the upper bound limit analysis method. A comparison of simulation results indicates that this technique can accurately identify the variation trend of basal stability against upheaval under the influence of isolation piles and achieve high calculation precision under the operating conditions of wide foundation pits and short isolation piles. Accordingly, a moderate increase in isolation pile parameters produces a significant supporting effect for narrowed foundation pits. Whereas for wide foundation pits, the supporting capacity of isolation piles can be maximized when pile length equals excavation depth.

Stability Analysis of the Foundation Pit and the Twin Shield Tunnels during Adjacent Construction

Due to the rapid development of urban rail transit and the development and use of underground space, foundation pit construction near existing subway tunnels is becoming increasingly common. When adjacent foundation pit projects are being constructed, it is difficult to avoid the influence on existing subway tunnels, which threatens the safe operation of subway tunnels. Thus, this study focused on the stability of the adjacent construction of foundation pits and twin shield tunnels. The finite element limit analysis (FELA) method was used to analyze the influences of construction scheme and relative position of the twin shield tunnels and adjacent foundation pit on the global stability of adjacent construction. Based on the research results, the guidance and design for the construction scheme and relative position are provided. The stability analysis on the foundation pit during the construction process is performed. The criteria of adjacent influential partition are proposed, and the influence of various input parameters (c/γD, φ, L/D and C/D) on the global stability of adjacent construction is discussed, respectively. A new design equation of safety factor of global stability of adjacent construction is obtained by curve fitting.

Stability analysis of nonhomogeneous and anisotropic stepped slopes under the influence of earthquakes

There have been many studies related to the stability of single-step slopes, but there have been few studies on the stability of stepped slopes. Based on the limit analysis method and the strength reduction method, the stability factor FS of a stepped slope in nonhomogeneous and anisotropic soils is calculated. Then, the calculation method in this paper is compared with previous studies to verify the correctness of the calculation method. Finally, the parametric analysis of the stepped slope is also carried out. The results show that the maximum error obtained by the calculation method of this paper does not exceed 5%, which verifies the rationality and effectiveness of the method in this paper. The ratio B/H of slope width to slope height significantly affects the stability of the slope. When B/H < 3, FS decreases sharply with increasing B/H; when B/H > 3, FS decreases slowly with increasing B/H. When the inclined angle, anisotropy parameter and seismic parameter of the slope increase, the stability of the stepped slope decreases; when the platform width parameter and soil nonhomogeneity parameter of the slope increase, the slope stability is also improved.

Limit Analysis for 3D Stability of Unsaturated Inhomogeneous Slopes Reinforced with Piles

Achieving slope stabilization by placing anti-slide piles has become an effective slope-reinforcement technique in geotechnical engineering. Based on the limit analysis method, this study investigated the 3D stability of unsaturated inhomogeneous piled slopes, incorporating three failure patterns: toe-failure, face-failure, and base-failure. The suction and effective unit weight profile under different effective saturation were considered and the soil cohesion was assumed as varied linearly with depth. A theoretical analysis of the lateral force acting on a row of rigid anti-slide piles with the same spacing in a row through plastically deforming soils is described while considering the soil suction and inhomogeneity. In addition, a new formula was developed to calculate the lateral forces provided by a row of piles. The genetic algorithm (GA) was used to obtain the minimum value of the factor of safety FS and the corresponding critical slip surface. The results of the study were compared with those of the existing literature to verify the feasibility of the proposed approach. In addition, the influences of key parameters on FS of 3D piled slopes were investigated by parametric analysis. The numerical results indicate that piled slope stability will be underestimated when the 3D effects and suction are not considered, and soil inhomogeneity has a negative impact on piled slope stability. It was also found that the probability for the occurrence of face-failure increases with increasing inhomogeneity and decreasing suction-induced effect, and the occurrence of face-failure significantly reduces the pile reinforcement effect. In addition, when the anti-piles are located at the slope toe, the base-failure mechanism yields the critical values of FS in most cases. This study can provide several reasonable suggestions for engineering applications and a theoretical basis for further studies on the stability of slopes reinforced with piles.

Influence of soft band on seismic slope stability by finite-element limit-analysis modelling

Soft bands may widely exist in sites, and their presence complicates the analysis of slope stability, particularly when the limit analysis method is adopted. Note that the use of the limit analysis method is capable of providing rigorous lower and upper bound solutions and hence is widely used in soil slope stability analyses. To account for soft band effects, finite-element lower-bound (FELB) and finite-element upper-bound (FEUB) approaches are introduced in this study. Finite element discretization aids in the construction of a statically allowable stress field and kinematically admissible velocity field, which are prerequisites for limit analysis. In combination with a modified pseudodynamic approach, seismic slope stability analysis is evaluated through linear programming. Lower and upper bound solutions of the slope safety factor (FoS) are optimized with an interior point algorithm, which aids in better estimating the true FoS of seismic slope stability. The nonassociated flow rule is also discussed herein. After validating the upper and lower linear programming models by limit equilibrium and FEM, the soft band’s effects on seismic slope stability are discussed, and a rotational-translational failure mechanism is revealed by critical velocity fields. A less than 5.1% difference between lower- and upper-bound solutions is favourable for providing a reliable reference for slope design.

Energy Response Characteristics of Laterally Loaded Piles

A bounding surface plasticity p − y model is used to examine the internal energy responses of a laterally loaded pile in static and dynamic simulations. The model response is first compared with those observed in physical centrifuge tests, including monotonic and cyclic loading, in a normally consolidated fine-grained soil. Then, the model is used to examine hypothetical static and dynamic situations. The monotonic load capacities determined by the model and an upper-bound limit analysis method are in agreement with that inferred from the physical test. The work done by external loads in static conditions is principally balanced by the plastic work in the p − y springs and the elastic strain energy in the structural pile elements, which are comparable in the cases examined. The numerical model is capable of reproducing accurately the observed test structural responses as well as the characteristics of plastic energy dissipation. In a dynamic free-vibration simulation, the pile is highly damped as a result of the significant plastic energy dissipation. The simple work-hardening elastic–plastic p − y model is theoretically sound, applicable to arbitrary loading histories, and requires the same level of effort to use as traditional p − y methods.

Three-dimensional seismic stability of locally loaded slopes under a rotational velocity field

In practical engineering, slopes subjected to local loads, like footings of buildings, are common. This paper aims to give an insight into the effect of seismic force on the stability of locally loaded slopes. Numerical methods can be used to study this problem, but they require much computational time. Contrarily, limit analysis method is an approach to perform slope stability analysis with high computational efficiency. Thus, an accurate approach in mechanical points is proposed for this problem based on limit analysis method herein. In the framework of limit analysis, existing research about this problem used a kinematically translational velocity field. However, the velocity field of the locally loaded slope at failure is proved to be rotational possibly. Thus, to fill this gap, a 3D rotational velocity field is employed herein to obtain limit loads on the slope top, which improves the existing upper-bound solutions obtained by using the translational velocity field. The particle swarm optimization algorithm and the Nelder-Mead simplex algorithm are employed to search the global minimum of the upper-bound estimation of the limit load. Parametric analysis is performed and it shows that the limit load increases with the increase of a/H or the internal friction angle φ but decreases as the slope angle β or the length-to-width ratio (L/t) of the local load increases. Furthermore, the limit load is found to decrease with the increase of the seismic coefficient kh and it is proportional to the seismic coefficient.

Analyses on face stability of shallow tunnel considering different constitutive models

Based on the finite element limit analysis method, the stability of the face in case of active failure under three constitutive models, the Mohr-Coulomb model (MC), the modified Cambridge model (MCC) and the Drucker-Prager model (DP), were analyzed. The ultimate support pressure of the face and the influence of factors such as different burial depth ratios (C/D), cohesion (c) and friction angle (φ) in the MC model are also discussed. The results show that the safety factor obtained by the MCC model under the same support pressure is always smaller than that of the MC model, and the difference is the largest when there is no support pressure. As the support pressure increases, it will gradually approach the MC model. When the support pressure is small, the safety factor obtained by the DP model is larger than the MC model, but when the support pressure is large, it is smaller than the MC model, and the final difference tends to be stable. It is necessary to select an appropriate constitutive model according to different rock masses in practical engineering. The self-stabilizing performance of the face is not affected by C/D, and the ultimate support pressure will increase with the increase of C/D, decrease linearly with the increase of cohesion, and decrease with the increase of friction angle. When the friction angle is small, the ultimate support pressure is greatly affected by C/D, and when the friction angle is large, it is hardly affected by C/D.

Fracturing and collapse behavior of masonry vaulted structures: a lattice-discrete approach

This study presents a novel meso-scale approach to investigate the fracturing and collapse behaviors of unreinforced masonry vaulted structures induced by spreading supports. Traditionally, the behavior of masonry vaulted structures is investigated by resorting on limit analysis method, that is a limited approach as: (i) it tackles just the failure condition of the arched structural configuration as it assumes the simultaneous formation of hinges once the thrust line reaches the edge of the masonry structure, (ii) it completely ignores the damage propagation phenomenon, starting from the trigger of the first fracture up to the complete structural failure condition. The comprehension of this fracturing process is fundamental aiming to analyze intermediate damage states for the check of serviceability limit states and to individuate a more realistic structural crack distribution in ultimate conditions. This paper proposes a thorough understanding of the fracturing behavior of masonry vaults based on non-linear fracture mechanics concepts. For this purpose, the Lattice Discrete Particle Model (LDPM) is adopted to simulate a variety of stone masonry vaulted structures up to their collapse. LDPM simulates the behavior of masonry at the stone level. The interaction between stones that are bounded by weak layers of mortar is governed by specific constitutive equations describing tensile fracturing with strain-softening, cohesive and frictional shearing, and compressive response with strain-hardening. The formation of hinges, the activation of the mechanism and the kinematic mechanism are analyzed for three different types of vaults, namely groin, barrel and depressed vaults, and for six different slenderness. The first conclusion of this study is that LDPM can be used as an alternative tool to perform typical limit analysis for the assessment of safety of arches and vaults in ultimate conditions. Most importantly, LDPM is able to show that the fracturing process is a progressive phenomenon, the cracked surfaces are never strictly symmetric respect to the vertical axis and do not appear simultaneously. In particular, the features of non simultaneity and asymmetry of the cracks increases as the distance between the crown of the vault and the imposts increases, i.e. going from depressed vaults to groin vaults. Finally, the evolution of the fracturing process occurs more progressively and exhibits less pronounced asymmetry in the case of depressed vaults as compared to groin vaults for which, in turn, the damage appears to be more brittle and characterized by asymmetry in the cracks distribution.

Characterizing model uncertainty of upper-bound limit analysis on slopes using 3D rotational failure mechanism

The slope stability assessment is a classical problem in geotechnical engineering. This topic have attracted many researcher’s attention and various theoretical models for predicting critical slope heights or safety factors in the light of the limit equilibrium (LE) method and the kinematical approach of limit analysis (LA) method. Meanwhile, a large number of experimental studies have been conducted to check the slope stability. Using centrifuge testing results, this paper aims to employ Bayesian method to characterize the model uncertainties of the classical three-dimensional rotational failure mechanism proposed by Michalowski and Drescher (2009) to predict critical slope heights in frictional soils, by incorporating the test uncertainties and parameter uncertainties. The obtained results show that the LA three-dimensional rotational failure mechanism overestimates the critical slope height compared with the LE method, and the experimental observational uncertainty has negligible influences on the posterior statistics of model uncertainty.

Sequential Plastic Method for 2D frames limit analysis

This work focuses on the limit analysis or plastic calculation of slender beam planar frames using a new method. Sufficient static equilibrium equations are proposed by the Principle of Virtual Displacements (PVD) and kinematic compatibility equations are proposed by the Principle of Virtual Forces (PVF). The load level is increased and the structure is solved step by step until the plastic collapse. Equilibrium equations are posed by virtual problems in displacements and compatibility equations are posed by simple virtual problems in equilibrium. The method provides the following results: the collapse load factor, the final collapse mechanism, the bending moments and the accumulated rotations in the plastic hinges in each load step of the structure. This method has advantages over the classical methods; first, with respect to the step-by-step method based on matrix formulation, especially in the case of beams and/or columns with uniform distributed load, and secondly, with respect to the kinematic direct method, since the sequential method provides more information on the quantities involved in the plasticization process of the structure and also goes directly to the calculation of the collapse mechanism without the need to test or combine possible mechanisms.

Effect of slope on lateral bearing capacity of nearshore large-diameter monopiles in cohesive soil

This study investigates the effect of slope on the lateral bearing capacity of nearshore large-diameter monopiles in cohesive soil by the finite element limit analysis (FELA) method. The reduction coefficient is introduced to quantify the capacity reduction due to the slope effect, and the critical distance making the slope-induced capacity reduction negligible is normalized for generalization. After the validations by cases of different soils, an alternative way to quickly and straightforwardly assess the pile bearing capacity in sloping stratum for the most frequently-encountered cases in engineering practice is provided by multiplying the reduction coefficients in several design tables with the capacity in level stratum. The subsequent parametric studies indicate that the reduction coefficient and critical dimensionless distance are mainly governed by the internal friction angle, slope angle, and pile embedded length-diameter ratio but are almost insensitive to dimensionless cohesion and slope height-pile embedded length ratio. In addition, the failure mode of the pile-slope system would develop from local failure to transitional failure and global failure with a significant drop in pile bearing capacity while the pile-slope parameters cannot guarantee slope stability.