Limit Analysis Approach Research Articles

A Genetic Algorithm NURBS-based new approach for fast kinematic limit analysis of masonry vaults

The present paper proposes a new Genetic Algorithm NURBS-based approach for the limit analysis of masonry vaults based on an upper bound formulation. A given masonry vault geometry can be represented by a NURBS (Non-Uniform Rational B-Spline) parametric surface and a NURBS mesh of the given surface can be generated. Each element of the mesh is a NURBS surface itself and can be idealized as a rigid body. An upper bound limit analysis formulation, which takes into account the main characteristics of masonry material is deduced, with internal dissipation allowed exclusively along element edges. The approach is capable of well predicting the load bearing capacity of any masonry vault of generic shape. It is proved that, even by using a mesh constituted by very few elements, a good estimate of the collapse load multiplier is obtained provided that the initial mesh is adjusted by means of a meta-heuristic approach (i.e. a Genetic Algorithm, GA) in order to enforce that element edges accurately represent the actual failure mechanism. The proposed method turns out to be both accurate and much less computationally expensive than existing methods for the limit analysis of masonry vaults.

A plane strain yield design approach to stability analysis of a column‐reinforced soil foundation under inclined loading

SummaryThe ultimate bearing capacity problem of column‐reinforced foundations under inclined loading is investigated within the framework of static and kinematic approaches of yield design theory. The configuration of a native soft clayey soil reinforced by either a group of purely cohesive columns (lime‐column technique) or a group of purely frictional columns (stone‐column technique) is analyzed under plane strain conditions. First, lower bound estimates are derived for the ultimate bearing capacity by considering statically admissible piecewise linear stress distributions that comply with the local strength conditions of the constitutive materials. The problem is then handled by means of the yield design kinematic approach of limit analysis through the implementation of several failure mechanisms, allowing the formulation of upper bound estimates for the ultimate bearing capacity. A series of finite element limit load solutions obtained from numerical elastoplastic simulations suggests that the predictions derived from the kinematic approach appear to be more accurate than the estimates obtained from the static approach. Comparison with available results obtained in the context of yield design homogenization demonstrates the accuracy of the proposed direct analysis, which may therefore be viewed as complementary approach to homogenization‐based approaches when a small number of columns is involved. Copyright © 2016 John Wiley & Sons, Ltd.

Effects of nonlinear strength parameters on stability of 3D soil slopes

Actual slope stability problems have three-dimensional (3D) characteristics and the soils of slopes have curved failure envelopes. This incorporates a power-law nonlinear failure criterion into the kinematic approach of limit analysis to conduct the evaluation of the stability of 3D slopes. A tangential technique is adopted to simplify the nonlinear failure criterion in the form of equivalent Mohr-Coulomb strength parameters. A class of 3D admissible rotational failure mechanisms is selected for soil slopes including three types of failure mechanisms: face failure, base failure, and toe failure. The upper-bound solutions and corresponding critical slip surfaces can be obtained by an efficient optimization method. The results indicate that the nonlinear parameters have significant influences on the assessment of slope stability, especially on the type of failure mechanism. The effects of nonlinear parameters appear to be pronounced for gentle slopes constrained to a narrow width. Compared with the solutions derived from plane-strain analysis, the 3D solutions are more sensitive to the values of nonlinear parameters.

M-N Interaction Effect on the Frames Failure Mechanisms

The collapse factor is a significant parameter in the framework of the safety assessment and economical design of ductile structures. This fact draws attention to the necessity of a careful assessment of the limit analysis approaches. The kinematics in these structures arises in fact from the actual rotation of the plastic hinges under axial force and bending moment. It can be shown that it is possible to obtain a reliable tool capable of competing with computationally expensive methodologies. The application of the methods of limit analysis involves a simplified and idealised model of the structure and, notwithstanding the fact that hundreds of papers have been devoted to the topic, some consequences of apparently unimportant simplifications still seem to have not been properly and firmly highlighted. This paper investigates the ultimate load and collapse modes of steel frames under combined vertical and horizontal forces through limit analysis.

Bearing capacity of non-associative coaxial granular materials by upper bound limit analysis and finite elements

ABSTRACTIn this paper, an upper bound estimate of the limit load on non-associative coaxial granular materials is presented. The kinematic approach of the upper bound limit analysis has been utilised. The failure mechanism is assumed to coincide with the direction of the shear bands at every point throughout the body. The shear band orientation in non-associative coaxial materials, i.e. those with the same major principal stress and major principal strain increment directions, can be found based on the angle of dilation and the major principal stress direction. Therefore, having known the stress field at limiting equilibrium, the orientation of the shear bands and hence, the failure mechanism can be obtained. In this study, the stress field is first determined by the method of stress characteristics. Then, the finite element interpolation technique is used to interpolate the stress field and to find the orientation of the shear bands at every point within the field. Once the failure mechanism and the stress state at every point along velocity discontinuities have been found, the upper bound limit analysis has been performed to estimate the limit load.

Required strength of geosynthetic in reinforced soil structures supporting spread footings in three dimensions

Little insight exists into the stability of reinforced slopes supporting spread footings. Existing research is largely focused on idealized, plane strain analyses that do not account for three-dimensional conditions. These conditions are significant because the geometries of both reinforced slopes and footings are three-dimensional in nature. In this study, the stability of reinforced slopes supporting spread footings is analyzed in three dimensions using a limit analysis approach and presented in a series of design charts. To better understand the impact of three-dimensional conditions, a parametric study is conducted to evaluate effects of various design parameters on required reinforcement strength.

Reliability-based assessment of existing masonry arch railway bridges

A great number of masonry arch bridges dates back to past centuries, being preserved by society due to their historical and still economic importance. Thereby, adequate preservation measures are required. Regarding masonry arch bridge’s structural condition, it is relevant to consider its age, and consequently deterioration, and the fact that these bridges are submitted to loads higher than those for which they were conceived, being imperative to assess their structural performance. Regarding safety assessment requirements, there are different reliability levels, whose objectives are to analyse the ultimate load-carrying capacity and the serviceability performance.This paper presents and discusses a framework that allows to determine the ultimate load-carrying capacity (Ultimate Limit State) of masonry arch bridges, using limit analysis and probabilistic approaches. Geometric and material data and load characterization, as well as inherent uncertainties will be also introduced. In order to determine the ultimate load-carrying capacity, the plastic theory will be employed, namely the limit analysis theorem, which is based on kinematic mechanisms. Since one of the main drawbacks of a probabilistic analysis is the required high computational resources, a sensitivity analysis is incorporated in order to reduce the analysis time. The presented framework is validated with an application to a set of existing Portuguese railway masonry arch bridges.

Analytical and Numerical funicular analysis by means of the Parametric Force Density Method

The funicular concept has often been used in different stages of structural analysis and design. This paper presents two new methods: Analytical , A-FDM , and Numerical method , N-FDM , based on a parametric application of the original Force Density Method (FDM ). This is an especially useful way of visualizing a set of solutions and optimizing, i.e. selecting one specific funicular related to a set of constraints. Two structural algorithms are implemented iteratively with Maple ® in real time, and output is also linked to AutoCAD ®. Maple ® facilitates control of geometrical constraints, while AutoCAD ® helps to show all parameterized data. Because of their practical interest, special emphasis is placed on masonry structures using a Limit Analysis approach and preliminary design. Examples of the application of both methods are depicted

Limit analysis of reinforced embankments on soft soil

Previous research into the stability of reinforced embankments founded on soft soil has presented limited studies based on a narrow range of assumed failure mechanisms. In this paper comprehensive parametric studies of reinforced and unreinforced embankments were conducted using the general purpose computational limit analysis approach Discontinuity layout optimisation (DLO). Comparisons with previous Limit Equilibrium and FE results in the literature showed good agreement, with the DLO analysis generally able to determine more critical failure mechanisms. Simplified, summary design envelopes are presented that allow critical heights and reinforcement strengths to be rapidly determined based on soft soil strength and depth, and show how the balance between soft soil strength and reinforcement strength combines to affect overall stability.

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation

The present two-part study aims at investigating the specific effects of Mohr–Coulomb matrix on the strength of ductile porous materials by using a kinematic limit analysis approach. While in the Part II, static and kinematic bounds are numerically derived and used for validation purpose, the present Part I focuses on the theoretical formulation of a macroscopic strength criterion for porous Mohr–Coulomb materials. To this end, we consider a hollow sphere model with a rigid perfectly plastic Mohr–Coulomb matrix, subjected to axisymmetric uniform strain rate boundary conditions. Taking advantage of an appropriate family of three-parameter trial velocity fields accounting for the specific plastic deformation mechanisms of the Mohr–Coulomb matrix, we then provide a solution of the constrained minimization problem required for the determination of the macroscopic dissipation function. The macroscopic strength criterion is then obtained by means of the Lagrangian method combined with Karush–Kuhn–Tucker conditions. After a careful analysis and discussion of the plastic admissibility condition associated to the Mohr–Coulomb criterion, the above procedure leads to a parametric closed-form expression of the macroscopic strength criterion. The latter explicitly shows a dependence on the three stress invariants. In the special case of a friction angle equal to zero, the established criterion reduced to recently available results for porous Tresca materials. Finally, both effects of matrix friction angle and porosity are briefly illustrated and, for completeness, the macroscopic plastic flow rule and the voids evolution law are fully furnished.

Limit analysis of 3D rock slope stability with non-linear failure criterion

The non-linear Hoek-Brown failure criterion has been widely accepted and applied to evaluate the stability of rock slopes under plane-strain conditions. This paper presents a kinematic approach of limit analysis to assessing the static and seismic stability of three-dimensional (3D) rock slopes using the generalized Hoek.Brown failure criterion. A tangential technique is employed to obtain the equivalent Mohr-Coulomb strength parameters of rock material from the generalized Hoek-Brown criterion. The least upper bounds to the stability number are obtained in an optimization procedure and presented in the form of graphs and tables for a wide range of parameters. The calculated results demonstrate the influences of 3D geometrical constraint, non-linear strength parameters and seismic acceleration on the stability number and equivalent strength parameters. The presented upper-bound solutions can be used for preliminary assessment on the 3D rock slope stability in design and assessing other solutions from the developing methods in the stability analysis of 3D rock slopes.

Kinematical Limit Analysis for a Slope Reinforced with One Row of Stabilizing Piles

Stabilizing piles are commonly used to improve slope stability. The load that a pile should sustain and the associated slip surface under a prescribed factor of safety are two critical design parameters in practice. Based on the kinematically admissible failure mechanism, a limit analysis approach is introduced, which considers two potential slip surfaces that individually lie in the upslope and downslope of the piles. The explicit formulas for the upslope thrust force and downslope resistance on a pile are derived, which enables the calculation of the maximum load on a pile and the corresponding critical slip surfaces using a computer program. The maximum load and critical slip surfaces under a design factor of safety are mainly affected by soil properties and five other parameters: location of the pile row, dip angle of the slope face, action point and dip angle of the force, and improvement coefficient of the slope stability. The relevant quantitative calculation can be performed using the proposed method. Moreover, the possible failure with the upslope soil surpassing the pile top under a specified design factor of safety is significantly illustrated in the paper, which is helpful in the rational determination of the location of the pile row.

Solutions for stability of anchored earth slopes subjected to seismic loading

The limit analysis approach is paid more attention to in the stability analysis of slopes subjected to seismic loading. However, the solutions obtained from the approaches are in general limited to a designated slope or the same kind of slopes. It seems that a wide applicability of the solutions in their present form is rather difficult. Based on the kinematic theorem and the assumption of combined log-spiral rotational failure mechanism, the integral formulation for expressing the rate of external work rate due to soil weight and seismic loading is derived under the framework of pseudo-static method considering the complex geometrical configurations and the multilayer soils in slopes with multi-row anchors. Further, the work-energy balance equation is established, and the analytical expressions for the global factors of safety (FS) and the yield acceleration coefficient of anchored slopes are obtained by using the strength reduction technique combined with the optimization method. Moreover, the permanent displacement with Newmark's method is also solved. Several representative examples are employed to validate the analytical solutions, and some comparisons of the FSs and the corresponding critical sliding surfaces obtained by various methods are also made. The results show that the analytical approach can yield the high-accuracy solutions and is widely applicable to the seismic stability analysis of complex slopes with multi-row anchors. Finally, a practical case is analyzed using the proposed approach. ©2016, Editorial Office of Chinese Journal of Geotechnical Engineering. All right reserved.

Face Stability Analysis for a Shield-Driven Tunnel in Anisotropic and Nonhomogeneous Soils by the Kinematical Approach

AbstractThe stability on a tunnel face is of great practical significance for safe construction in tunnel engineering. In this work, an advanced three-dimensional (3D) failure mechanism is extended to the face stability analysis of a tunnel driven in anisotropic and nonhomogeneous soils using the kinematic approach of limit analysis. The results obtained from the presented approach are in good agreement with the existing ones, showing that this method can be further applied to the proposed situation in this paper. Two common examples of cohesion variation, a linear variation with depth and a layered soil, are then analyzed. The numerical results show that both anisotropy and nonhomogeneity have a significant impact on the critical face pressure, especially when the cohesion is relatively large or when there is a weaker layer. Therefore, these two factors should be taken into account in tunnel design.

An analytical Lode angle dependent damage model for ductile porous materials

We propose a statical limit analysis approach for ductile porous materials, based on a general three dimensional (3D) trial stress field. This approach delivers a new macroscopic criterion which explicitly accounts for stress Lode angle effects. After illustrating the yield surfaces, we provide the macroscopic plastic flow rule and void growth equations. Based on the above results, a full 3D constitutive damage law is formulated and numerically implemented with an isotropic hardening law. This allows to predict the macroscopic response of the voided material under arbitrary loadings. Finally, the influence of the stress Lode angle as well as that of the stress triaxiality are fully illustrated.

Effects of nonlinear failure criterion on the three-dimensional stability analysis of uniform slopes

This paper presents a kinematic approach of limit analysis to evaluate the effects of nonlinear failure criterion on the three-dimensional (3D) stability analysis of uniform slopes. For slopes in various clays, the upper bounds on the critical heights associated with the linear and nonlinear failure criteria are obtained and comparisons of the linear and nonlinear results are made to investigate their differences. The differences between the linear and nonlinear solutions change with varying slope inclination, resulting in two critical values on the slope inclination angle. In the intermediate range of slope inclinations, the critical heights associated with the linear failure criterion are smaller than the nonlinear results, and the underestimation by using the linear failure criterion cannot be neglected. For other cases, the analyses based on the linear failure criterion will overestimate the stability of actual slopes obeying the nonlinear failure criterion. Meanwhile, the 3D effects have insignificant influences on the two critical slope inclination angles. However, the sensitivity of the critical heights to the nonlinearity of soil strength envelope appears to be more pronounced for slopes with a smaller width.

Interval Limit Analysis Within a Scaled Boundary Element Framework

The paper proposes a novel approach for the interval limit analysis of rigid-perfectly plastic structures with (nonprobabilistic) uncertain but bounded forces and yield capacities that vary within given continuous ranges. The discrete model is constructed within a polygon-scaled boundary finite element framework, which advantageously provides coarse mesh accuracy even in the presence of stress singularities and complex geometry. The interval analysis proposed is based on a so-called convex model for the direct determination of both maximum and minimum collapse load limits of the structures involved. The formulation for this interval limit analysis takes the form of a pair of optimization problems, known as linear programs with interval coefficients (LPICs). This paper proposes a robust and efficient reformulation of the original LPICs into standard nonlinear programming (NLP) problems with bounded constraints that can be solved using any NLP code. The proposed NLP approach can capture, within a single step, the maximum collapse load limit in one case and the minimum collapse load limit in the other, and thus eliminates the need for any combinatorial search schemes.

Lower Bound Axisymmetric Limit Analysis Using Drucker–Prager Yield Cone and Simulation with Mohr–Coulomb Pyramid

This paper presents a lower bound limit analysis approach for solving an axisymmetric stability problem by using the Drucker–Prager (D–P) yield cone in conjunction with finite elements and nonlinear optimization. In principal stress space, the tip of the yield cone has been smoothened by applying the hyperbolic approximation. The nonlinear optimization has been performed by employing an interior point method based on the logarithmic barrier function. A new proposal has also been given to simulate the D–P yield cone with the Mohr–Coulomb hexagonal yield pyramid. For the sake of illustration, bearing capacity factors Nc, Nqand Nγhave been computed, as a function of ϕ, both for smooth and rough circular foundations. The results obtained from the analysis compare quite well with the solutions reported from literature.

Three-dimensional reinforced slopes: Evaluation of required reinforcement strength and embedment length using limit analysis

Virtually all analyses of reinforced earth structures are based on two-dimensional (2D) conditions, ignoring three-dimensional (3D) effects despite adhering to three-dimensional behavior in reality. In this study, the three-dimensional effects of three-dimensional conditions on reinforced earth structure stability are considered, and employed to determine the required strength and length of reinforcement using a Limit Analysis approach. The results of the three-dimensional stability analysis are compared with idealized two-dimensional results and examples from prior literature. As expected, use of two-dimensional stability analyses for reinforced soil slopes yield results that are more conservative than those considering three-dimensional effects. The effects of various parameters on the three-dimensional solution are investigated and design charts are proposed. The solutions demonstrate that for practical scenarios, three-dimensional effects may affect the required length of reinforcement more than the associated required tensile strength.

Analytical solutions to assess the stability of rock slopes subject to cracks via limit analysis

Based on the kinematic approach of limit analysis, a full set of upper bound solutions for the stability of homogeneous rock slopes subjected to cracks are obtained. The generalized Hoek-Brown failure criterion is adopted to describe the non-linear strength envelope of rocks. In this paper, critical failure mechanisms are determined for cracks of known depth but unspecified location, cracks of known location but unknown depth, and cracks of unspecified location and depth. It is shown that there is a nearly up to 50% drop in terms of the stability factors for rock slopes intersected by cracks compared with intact ones. Tables and charts of solutions in dimensionless forms are presented for ease of use by practitioners.