Limit Analysis Approach Research Articles

Investigation of the stability of submarine sensitive clay slopes underwave-induced pressure

The exploration and exploitation of marine georesources ordinarily disturbs the submarine soft clay surrounding construction areas and leads to a significant decrease in the shear strength of structured and sensitive clayey soils in submarine slopes. Under wave action, local slides can even trigger large-scale submarine landslides, which pose a serious threat to offshore infrastructure such as pipelines and footings. Therefore, accurately evaluating the stability of submarine sensitive clay slopes under wave-induced pressure is one of the core issues of marine geotechnical engineering. In this paper, a kinematic approach of limit analysis combined with strength reduction technique is presented to accurately evaluate the real-time stability of submarine sensitive clay slopes based on the log-spiral failure mechanism, where external work rates produced by wave-induced pressure on slopes are obtained by the numerical integration technique and then are applied to the work-energy balance equations. The mathematical optimization method is employed to achieve the safety factors and the critical sliding surfaces of submarine slopes at different time in a wave cycle. On this basis, the stability of submarine sensitive clay slopes under various wave parameters is systematically investigated. In particular, extreme wave conditions and special cases of slope lengths no more than one wavelength are also discussed. The results indicate that waves have some negative effects on the stability of submarine sensitive clay slopes.

Predicting seismic permanent displacement of soil walls under surcharge based on limit analysis approach

Seismic permanent displacement of the soil walls plays an important role in design of these structures. Due to the increase in growth of urban areas and the limitations in use of flat grounds, many structures are built near slopes and retaining walls. During earthquakes, these structures can apply an additional surcharge on the wall. The intensity and location of the surcharge is of considerable importance on the seismic displacements of the soil wall. In this study, by using the limit analysis and upper bound theorem, seismic permanent displacement of the soil wall under surcharge has been analyzed. Thus, a formulation is presented for calculating the yield acceleration and seismic displacement for different surcharge conditions. The effect of seismic acceleration, surcharge intensity, its location and soil properties is investigated. A parameter called the “displacement coefficient” is proposed, and is a potential modification for Newmark’s sliding-block method.

Formulating the in-plane frictional resistances and collapse mechanisms for multi-storey masonry block walls

In this paper a macro-block model accounting for frictional resistances is presented to assess the lateral strength of a multi-storey masonry block wall. The kinematic approach of limit analysis is used to define the load factor causing the onset of rocking-sliding mechanism under in-plane horizontal loading. A dry frictional contact condition is assumed at the rigid block interfaces, according to the Coulomb's law with non-associated flow rule. The key aspect of the proposed approach is the introduction of a criterion to evaluate the contribution of the actual frictional resistances depending on the inclination angle of the crack line. An accurate assessment of the frictional resistances is also obtained by distinguishing two different contributions (the wall own weight and additional vertical loads) and their application points. Hence, a sensitivity analysis is performed with respect to the overloading condition, the friction coefficient, and geometrical parameters such as the shape ratios of the wall and the unit block and the number of rows. The analytical results of the proposed model are also validated against results from other existing macro and micro-block modelling approaches in terms of load factor. The comparison confirms the reliability of the proposed model that allows, with similar results, great simplification of the computational effort with respect to micro-block models.

Face stability of a tunnel excavated in saturated nonhomogeneous soils

Most subways excavated in nonhomogeneous and water-bearing soils but few existing analytical models focus on the combined effect of soil nonhomogeneity and pore water pressures. To address this issue, a three-dimensional (3D) rotational collapse mechanism is improved to investigate the effect of vertical variability of soil strength on the face stability of a circular tunnel excavated in saturated nonhomogeneous soils by means of the kinematical approach of limit analysis. The pore water pressure distribution obtained numerically is interpolated on the 3D improved collapse mechanism by a linear interpolation technology. The effectiveness of the developed model is verified by comparing the required face pressures with the existing solutions, the numerical simulations and the monitoring data of the Shenzhen urban rail transit line 9 project. The effects of nonhomogeneous friction angles and cohesion and water table elevation on the face stability are investigated.

Required unfactored geosynthetic strength of three-dimensional reinforced soil structures comprised of cohesive backfills

Conservative design of Geosynthetic-reinforced soil structures (GRSSs) is commonly limited to two-dimensional (2D) conditions, ignoring the influence of possible cohesion in backfill material. However, the actual stability of GRSSs is directly influenced by the presence of cohesion – true or apparent – in backfill as well as three-dimensional (3D) effects. In this study, a 3D rational failure mechanism based on the kinematic approach of limit analysis is adopted to assess the stability of GRSSs comprised of cohesive backfills. Within this study, the influence of 3D effects, varying pore water pressures, varying backfill cohesion, and a range of slopes on long-term stability are illustrated in a series of convenient design charts. The results of 3D stability analyses for geosynthetic reinforced walls constructed with cohesive backfills are compared with the results obtained from design guidelines. As expected, when GRSSs are well-drained and relatively narrow in width - or when increasing levels of cohesion are present in the backfill - more stable conditions are realized. For practical scenarios, however, it is critical that cohesive soils should be utilized as backfill with great caution and reliable drainage conditions. Nonetheless, the presented solutions are directly useful towards the assessment of failures of real GRSSs, as they may be constructed with marginal fills that exhibit cohesion, accumulate pore water pressure and often exhibit failure conditions that are three-dimensional in nature.

Intricacies in three‐dimensional limit analysis of earth slopes

SummaryAnalyses of three‐dimensional slope failures can be elaborate because of the complexity owed to the geometry of the failure mechanism that needs to conform to an admissible kinematics of the slope collapse. This admissibility is imposed by the soil limit state condition, the normality plastic flow rule, and the boundary conditions. The kinematic approach of limit analysis is employed, and a rotational 3D slope failure is revisited. The study leads to the conclusion that some geometric constraints used in previous studies limit the range of admissible mechanisms resulting in overestimating stability factors. A set of results is presented that was obtained using an algorithm that allowed eliminating limitations present in previous studies. The largest improvements in the solutions were found for undrained failures of narrow slopes. For a 30° slope limited in width by the width‐to‐height ratio of 0.6, the stability factor calculated in previous studies overestimated the current calculations by nearly 39%. This overestimation is smaller for drained failures, and it drops significantly with an increase in the width of the failure mechanism.

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.

Modelling and analysis of South Indian temple structures under earthquake loading

The gopuram (multi-tiered entrance gateway) and the mandapam (pillared multi-purpose hall) are two representative structural forms of South Indian temples. Modelling and seismic analysis of a typical 9-tier gopuram and, 4- and 16-pillared mandapam of the 16th century AD Ekambareswar Temple in Kancheepuram, South India, are discussed. The seismic input is based on a probabilistic seismic hazard analysis of the archaeological site. Two modelling strategies, namely lumped plasticity and distributed plasticity modelling, and three analysis approaches, namely linear dynamic, non-linear, static and dynamic analyses were adopted for the seismic assessment of the gopuram. Unlike slender masonry towers, the vulnerable part of the gopuram could be at the upper levels, which is attributable to higher mode effects, and reduction in cross section and axial stresses. Finite element and limit analysis approaches were adopted for the assessment of the mandapam. Potential collapse mechanisms were identified, and the governing collapse of lateral load, calculated based on limit theory, was compared with the seismic demand as a safety check. Simple relations, as a means of rapid preliminary seismic assessment, are proposed for the mandapam.

Face stability analysis of large-diameter slurry shield-driven tunnels with linearly increasing undrained strength

The kinematic approach of limit analysis is explored in three-dimensional face stability of large-diameter tunnels in soils with linearly increasing undrained strength with depth. Due to the slurry density by a shield machine, the support pressure is not uniformly distributed on the face. Such a non-uniform distribution cannot be ignored, especially for the large-diameter shield tunnels. This paper includes it into the stability analysis of the tunnel face subjected to local and global failures. The failure mechanism with a spherical cap is adopted to obtain the least upper-bound solutions of local stability of the tunnel face. To evaluate its global stability, a continuous velocity field with a toric envelope is employed and yields the critical pressures on tunnel face against collapse and blow-out. The calculated results are compared with the solutions derived from other well-established methods for verification of the presented approach. An approximation formula based on the derived upper-bound solutions, is given to directly calculate the necessary collapse and blow-out pressures, which can be used for preliminary design in practice. An example is given to illustrate its convenient use.

Stability analysis of leaning historic masonry structures

This paper introduces an automatic, powerful and easy to use procedure for undertaking stability analyses of leaning historic masonry structures, based on an upper bound finite element limit analysis (FELA) approach. The procedure proposed here consists of a comprehensive workflow which involves the automatic point cloud manipulation, the 3D mesh generation of the actual geometry for structural purposes (e.g. FE mesh), and a two-step FELA that reduces drastically optimization variables assuming only active few elements inside a restricted processing zone. To generalize the Heyman's intuition to complex real geometries, the use of a 3D upper bound FELA with a recursive kernel of variables reduction becomes necessary for a precise evaluation of the limit inclination that makes the structure collapse under gravity loads. This outcome permits to estimate the structural health condition of a historic structure by comparing the critical inclination angle against the actual one. To demonstrate the effectiveness of the automated procedure, the southwest leaning tower of the Caerphilly castle (Wales, UK) is investigated and failure mechanisms with collapse inclination angles are evaluated through FELA. The proposed procedure presents a high degree of automation at each operational level and, hence, could be effectively used to assess the stability of historic structures at a national scale and provide useful information to asset owners to classify the structural health condition of leaning historic masonry structures in their care.

Three-dimensional stability analysis of slope in unsaturated soils considering strength nonlinearity under water drawdown

In slope stability analyses, the soil has been frequently considered dry or saturated, whereas the soil is in fact unsaturated in many cases. Owing to the existence of matric suction, the strength properties of unsaturated soils are greatly different from dry or saturated soil, and thereby leading to diverse stability conditions of the slope. Based on the kinematical approach of limit analysis, in this study, a three-dimensional (3D) stability analysis was conducted for slopes in unsaturated soils. The effects of matric suction distribution patterns, the nonlinearity of the strength under soil-water characteristic curves and the water drawdown were investigated to explore the shear strength and stability of a slope in unsaturated soil. It was found that, when the matric suction is uniformly distributed, the stability factors of the slope increase linearly along with the matric suction. In addition, when the matric suction increases linearly with the depth, the enhancing effect of the matric suction on the slope stability intensifies as the variation magnitude of the matric suction increases, especially for a gentle slope with a narrower width. For nonlinear patterns, different soil-water characteristic curve (SWCC) models will lead to diverse shear strengths and stability conditions of the slope. The water drawdown has significant effects on the functions of the matric suction as well as slope stability.

Corner failure in masonry buildings: An updated macro-modeling approach with frictional resistances

Abstract The failure mode of a free corner in masonry buildings , still poorly investigated, is one of the most common failure mechanisms occurring and clearly recognizable in the aftermath of a seismic event. It is characterized by the formation of a masonry wedge, mainly due to the thrust of roof elements in addition to inertial forces , and it generally involves rocking-sliding motions along the cracks on the interlocked orthogonal walls. The onset of this failure mode is herein analyzed by means of an upgraded macro-block model, based on the kinematic approach of limit analysis and accounting for the influence of frictional resistances on the collapse load multiplier and the related crack pattern. An original criterion weighting the role of rocking vs. sliding motion on the collapse load factor is developed and a formulation with general applicability is obtained. Several parametric analyses are performed in order to highlight the influence of geometrical, mechanical and loading parameters on the seismic capacity of the corner. The reliability of the proposed model and solution procedure is confirmed through the comparison with the results provided by other macro-block models existing in the literature. The final perspective is the next implementation of the proposed model in FaMIVE (Failure Mechanism Identification and Vulnerability Evaluation) applicative.

Upper bound solutions for face stability of circular tunnels in non-homogeneous and anisotropic clays

This paper presents a three-dimensional stability analysis of a circular tunnel face in non-homogeneous and anisotropic undrained clay using the kinematic approach (upper bound) of limit analysis. The proposed failure mechanism consists of a cylindrical rigid block and a toroidal shear zone with variable radius, and the closed-form analytical expressions of velocity field are derived within the framework of an orthogonal curvilinear coordinate system. The critical collapse-supporting pressure and stability ratio are obtained through optimization with respect to the geometrical parameters of the mechanism. Two types of non-homogeneous undrained strength, linearly changing with depth, and two-layer clays with constant undrained strength are investigated. Meanwhile, the 3-D finite element analysis is employed to validate the proposed failure mechanism. The upper bound solutions of the stability ratio of the proposed mechanism compare reasonably well with the upper bound solutions from the finite element analysis and show significant improvements over the existing upper bound solutions in single layer clays with homogeneous and isotropic undrained strength. The results show that the critical collapse pressure decreases with the increase in the non-homogeneous ratio and the anisotropic ratio and increases with the ratio between the undrained strength of the top layer and of the bottom layer.

Upper bound analysis of 3D static and seismic active earth pressure

This paper adopts a three-dimensional (3D) rotational failure mechanism to calculate the static and seismic active earth pressure acting on rigid retaining walls within the framework of the kinematic approach of limit analysis. The failure mechanism can be generated through rotating a varying circle defined by two log-spirals, and a plane strain block is inserted into the mechanism for consideration of retaining walls with different widths. Quasi-static representation using the seismic coefficient concept is applied to consider horizontal earthquake effects for the seismic active earth pressure calculation. Two cases with the different points of action of resultant earth pressure are included in this study. In the realm of soil plasticity, the new expressions about 3D active earth pressure are educed from the work-energy balance equation. The critical solution is obtained from an optimization scheme where the maximum of active earth pressure is obtained. Numerical results for different parameters are calculated and presented in the forms of graphs and tables, which may serve as a useful tool for practical application. Compared with existing 2D and 3D solutions, the validity of the proposed method is shown. A parametric study is conducted to investigate the effects of different parameters on the static and seismic active earth pressure under 3D conditions.

A kinematic limit analysis approach for seismic retrofitting of masonry towers through steel tie-rods

The paper provides an insight into the seismic strengthening of masonry towers by means of horizontal and vertical steel tie-rods. The approach is based on the kinematic theorem of limit analysis with pre-assigned failure mechanisms. Among all the possible ones, five of them commonly observed during post-earthquake surveys are selected: (#1) vertical splitting; (#2) base rocking; (#3) Heyman’s diagonal rocking; (#4) combination of splitting and diagonal rocking; (#5) base sliding.The aim is to put at disposal a procedure that can be used in any case of technical interest. To provide general output applicable in different contexts, towers are supposed isolated and idealized with a constant hollow square cross-section, without openings and any type of irregularity. Different mechanisms can be activated as a consequence of geometric features (base, height and thickness of the walls) and masonry mechanical properties, here assumed obeying a Mohr-Coulomb failure criterion.Thanks to the simplicity of the approach, comprehensive sensitivity analyses with different heights, base widths and wall thicknesses varying in the range of technical interest, as well as large scale Monte Carlo (MC) simulations with several geometries and three different sets of mechanical properties are carried out.The possible introduction of horizontal and vertical steel tie-rods is investigated in the same way, simply considering the contribution of the reinforcement in the internal dissipation in limit analysis computations. The results of the analyses show that, depending on the geometry of the tower and the mechanical properties of masonry, different mechanisms can be activated and therefore the choice of the reinforcement must be done on the basis of the expected failure mode. In addition, it is possible to predict the change in the active failure mechanism due to the introduction of reinforcement, as well as to evaluate the increase in the load carrying capacity.

Upper-bound solutions for face stability of circular tunnels in undrained clays

The aim of this paper is to determine the critical supporting pressure on the circular face of an advancing tunnel in undrained clay in which undrained strength increases linearly with depth. Based on the kinematic approach of limit analysis, a continuous velocity field with a toric envelope is adopted to yield upper bounds of the face pressure for both collapse and blow-out. Combining with a closed-form expression describing the overall velocity field, an optimisation procedure is carried out to find the lowest upper-bound solutions. The calculated results are presented in terms of a dimensionless coefficient N (called the stability number) and then stability charts for use are produced. Alternatively, an approximation formula is given to estimate easily the collapse and blow-out pressure in practice.

Strength predictions of clear wood at multiple scales using numerical limit analysis approaches

This work aims at a new approach for understanding failure mechanisms and predicting wood strengths, which are strongly influenced by the complex hierarchical material system of wood. Thus, a mechanical concept, where different microstructural characteristics are incorporated, appears to be necessary, based on the division of wood into meaningful scales of observation. At each scale, effective strength properties are to be determined and a multiscale approach needs to be applied, for which conventional numerical methods appear to be inefficient. In this work, numerical limit analysis approaches are further developed and applied for the first time to wood, complementing conventional methods successfully at certain scales of observation in a multiscale ‘damage’ approach.Limit analysis belongs to the group of direct plastic analysis methods, focusing exclusively on the time instant of structural collapse, and delivering the ultimate strength. Compared with conventional numerical approaches that have previously been applied to wood, limit analysis approaches are much more stable and efficient.In this work, orthotropic failure criteria and periodic boundary conditions are implemented into both lower bound and upper bound numerical limit analysis formulations. As numerical results, effective failure surfaces are obtained at both annual ring scale and clear wood scale. A validation at clear wood scale indicates that this new approach is very promising.

Evaluation of the capacity surfaces of reinforced concrete sections: Eurocode versus a plasticity-based approach

The classical Eurocode-compliant ultimate limit state (ULS) analysis of reinforced concrete sections is investigated in the paper with the aim of verifying if and how this well-established design procedure can be related to plasticity theory. For this reason, a comparative analysis concerning capacity surfaces of reinforced concrete cross sections, computed via a ULS procedure and a limit analysis approach, is presented. To this end, a preliminary qualitative discussion outlines modeling assumptions aiming to reproduce the physical behavior of reinforced concrete cross sections with respect to ductility and confinement issues. Besides the theoretical importance of the proposed approach, numerical experiments prove that limit analysis yields not only very accurate results but also a computationally effective procedure that can be affordably used in common design practice.

Three dimensional face stability of a tunnel in weak rock masses subjected to seepage forces

Reliable prediction of tunnel face stability is a key challenge for tunnel engineering, especially when drilling in highly fractured rock masses under the water table. This work aims to study face stability of a circular tunnel in weak rock masses under the water table based on an advanced three-dimensional (3D) rotational collapse mechanism in the context of the kinematical approach of limit analysis. The fractured rock masses are characterized by the Hoek-Brown failure criterion. A 3D steady-state seepage field obtained numerically is used to interpolate hydraulic heads of the 3D collapse mechanism and seepage force is incorporated into this predicting model by directly considering it as a body force. The results provided by the presented approach are compared with those of numerical calculations, showing a good agreement. The proposed work also provides an improvement with respect to other existing solutions. Thanks to the high computational efficiency of the presented method, four sets of normalized charts are obtained for a tunnel driven in weak rock masses under the water table.

Analytical representation of Mohr failure envelope approximating the generalized Hoek-Brown failure criterion

The latest version of the Hoek-Brown criterion, referred to as the generalized Hoek-Brown (GHB) criterion, is an empirical failure condition for rock mass that correlates the rock mass strength with the data collected from the laboratory tests and field observations by use of GSI values. A shortcoming of the GHB criterion is that it does not provide an explicit relation between the shear and normal stresses on the failure plane, which limits its application in many geotechnical design calculations. In order to extend the application of the GHB failure criterion to limit equilibrium analysis and/or the upper-bound approach of limit analysis, the analytical approximations of the Mohr failure envelope corresponding the GHB criterion are derived. At present, the exact formulation is not available except for a=0.5, i.e. for a rock mass with GSI=100. In the formulation proposed here, the 1st- and 2nd- order Taylor approximations are employed to define the relation between the dimensionless normal stress and the sine of the instantaneous friction angle. Given the latter, the corresponding shear stress at failure is calculated from the established explicit relationship between the shear stress and the instantaneous friction angle. The Taylor expansion is carried out by taking the sine of the instantaneous friction angle as the expansion variable. A series of examples are given which demonstrate that the approximated Mohr envelope, derived by invoking the 2nd-order Taylor approximation with the expansion center of 0.5, is nearly identical to the exact analytical form. Even the 1st-order Taylor approximation with the expansion center of 0.5 is proven to be satisfactory. Furthermore, it is shown that the formulation is valid for a broad range of values of GSI. The latter has been verified by simulating the conventional triaxial compression tests within the framework of the critical plane approach (CPA) adopting the proposed form of Mohr failure envelope as a failure function. The results confirm that the calculated axial stresses at failure and the orientation of the failure plane are almost identical to the values stipulated by the original GHB criterion.