Limit Analysis Formulation Research Articles

Upper-Bound Axisymmetric Limit Analysis Using the Mohr-Coulomb Yield Criterion, Finite Elements, and Linear Optimization

An upper-bound limit analysis formulation has been presented for solving an axisymmetric geomechanics stability problem using the Mohr-Coulomb failure criterion in conjunction with finite elements and linear programming. The method is based on the application of the von Karman hypothesis, and it requires only nodal velocities as the basic unknown variables. The computational effort needed to solve the axisymmetric problem becomes almost the same as that required for an equivalent plane strain case. By using the proposed method, bearing capacity factors were obtained for a circular footing placed over a cohesive-frictional soil medium. Nodal velocity patterns were also examined. Necessary comparisons have also been given to examine the usefulness of the proposed formulation.

Application of discontinuity layout optimization to three-dimensional plasticity problems

A new three-dimensional limit analysis formulation that uses the recently developed discontinuity layout optimization (DLO) procedure is described. With DLO, limit analysis problems are formulated purely in terms of discontinuities, which take the form of polygons when three-dimensional problems are involved. Efficient second-order cone programming techniques can be used to obtain solutions for problems involving Tresca and Mohr–Coulomb yield criteria. This allows traditional ‘upper bound’ translational collapse mechanisms to be identified automatically. A number of simple benchmark problems are considered, demonstrating that good results can be obtained even when coarse numerical discretizations are employed.

Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects

In this paper, a formulation for limit analysis of three-dimensional masonry structures discretized as rigid block assemblages interacting through no-tension and frictional contact interfaces is developed. Linear and piecewise linearized yield functions are used for rocking, sliding and torsion failure. A simple yield condition has been defined to take into account interaction effects of shear force with torsion and bending moment. Associative flow rules are considered for strain rates. On the basis of the developed governing equations, the limit analysis problem has been formulated as a nonlinear mathematical program. An iterative solution procedure based on linear programming is used to solve the limit analysis problem and to take into account nonlinearities due to the influence of bending moments and shear stresses on torsion strength. The results of experimental investigations on out-of-plane masonry walls constrained at one edge and different examples from literature were considered for validation. Comparison with existing formulations is carried out.

Exact solutions of sequential limit analysis of pressurized cylinders with combined hardening based on a generalized Hölder inequality: Formulation and validation

The paper aims to investigate nonlinear combined isotropic/kinematic hardening cylinders under internal proportional pressure by sequential limit analysis. The Armstrong–Frederick kinematic hardening model is adopted and the Voce hardening law is incorporated for isotropic hardening behavior. In particular, we establish the kinematic formulation of sequential limit analysis from the corresponding static formulation by a generalized Hölder inequality. Especially, it is found that the derived kinematic formulation involving combined isotropic/kinematic hardening is equivalent to that by the bipotential concept. Further, exact solutions of plastic limit pressure were developed analytically by performing both static and kinematic limit analysis. Finally, the problem formulation and the solution derivations presented here are validated by a very good agreement between the numerical result of exact solutions of the present work and the upper bounds from the kinematic formulation available in literature.

A locking-free stabilized kinematic EFG model for plane strain limit analysis

An element free Galerkin (EFG) based formulation for limit analysis of rigid-perfectly plastic plane strain problems is presented. In the paper it is demonstrated that volumetric locking and instability problems can be avoided by using a stabilized conforming nodal integration scheme. Furthermore, the stabilized EFG method described allows stable and accurate solutions to be obtained with minimal computational effort. The discrete kinematic formulation is cast in the form of a second-order cone problem, allowing efficient interior-point solvers to be used to obtain solutions. Finally, the performance of the stabilized EFG method is illustrated by considering several numerical examples.

A novel augmented Lagrangian‐based formulation for upper‐bound limit analysis

SUMMARYThis paper describes a novel upper‐bound formulation of limit analysis. This formulation is an innovative variant of an existing two‐field mixed formulation based on the augmented Lagrangian method also developed by the authors. A natural approach is used to describe the deformation of each finite element. Furthermore, and in contrast to the previous formulation, two independent field approximations are now both used to define the velocity field, defined globally and at element level. It is shown that this feature allows a governing system of uncoupled linear equations to be obtained. Some numerical examples in plane strain conditions are presented in order to illustrate the current model performance. In conclusion, the potential and advantages of this new approach are discussed. Copyright © 2011 John Wiley & Sons, Ltd.

Stability numbers for an unsupported vertical circular excavation in c– ϕ soil

By using an axisymmetric lower bound finite element limit analysis formulation, the stability numbers ( γH/ c) for an unsupported vertical circular excavation in a cohesive–frictional soil have been generated. The numerical results are obtained for values of normalized excavation height ( H/ b) and friction angle ( ϕ) greater than those considered previously in the literature. The results compare well with those available in literature. The stability numbers presented in this note would be beneficial from a design point of view.

A novel augmented Lagrangian approach for Limit Analysis computations

In this paper, we introduce a novel upper bound formulation of Limit Analysis applied to plane strain problems. This formulation can considered to be a variant of an existing two-field mixed formulation based on the augmented Lagrangian method also developed by the authors. In contrast to the previous formulation, two independent field approximations are now used to define the velocity field, globally and at element level. It is shown that this feature allows a governing system of uncoupled linear equations to be obtained. Our main concern is to achieve a formulation with appropriate characteristics for implementation in an efficient parallel algorithm. Some numerical examples are presented in order to illustrate the current model performance.

Stability of an unsupported vertical circular excavation in clays under undrained condition

By using the axisymmetric finite elements static limit analysis formulation, proposed recently by the authors, the stability numbers ( γH/ c o ) for an unsupported vertical circular excavation in clays, whose cohesion increases with depth, have been determined under undrained condition; γ = unit weight, H = height of the excavation and c o = cohesion along ground surface. The results are obtained for various values of H/ b and m; where b = the radius of the excavation and m = a non-dimensional parameter which accounts for the rate of the increase of cohesion with depth. The values of the stability numbers increase continuously both with increases in H/ b and m. The results obtained in this study compare well with those available in literature.

Bearing capacity factor Nγfor a rough conical footing

The bearing capacity factor Ny is computed for a rough conical footing placed over horizontal ground surface. The axisymmetric lower bound limit analysis formulation, in combination with finite elements and linear programming, proposed recently by the authors is used in this study. The variation of Ny with cone apex angle (B), in a range of 30o-180o, is obtained for different values of ?; where ? is soil friction angle. For ? 30o, the minimum magnitude of Ny is found to occur generally between B = 120o and B = 150o. In all the cases, it is noticed that the magnitude of Ny becomes maximum for B = 30o. For a given diameter of the cone, the area of the plastic zone reduces generally with an increase in B. The obtained values of Ny are found to compare quite well with those available in literature.

Vertical uplift resistance of circular plate anchors in clays under undrained condition

The vertical uplift resistance of circular plate anchors, embedded horizontally in a clayey stratum whose cohesion increases linearly with depth, has been obtained under undrained ( ϕ = 0) condition. The axi-symmetric static limit analysis formulation in combination with finite elements proposed recently by the authors has been employed. The variation of the uplift factor ( F c ) with changes in the embedment ratio ( H/ B) has been computed for several rates of increases of soil cohesion with depth. It is noted that in all the cases, the magnitude of F c increases continuously with depth up to a certain value of H cr/ B beyond which the uplift factor becomes essentially constant. The proposed static limit analysis formulation is seen to provide acceptable results even for the two other simple chosen axi-symmetric problems.

Dynamic limit analysis formulation for impact simulation of structural members

This paper introduces an extended concept of limit analysis to deal with the dynamic equilibrium condition considering the inertia and strain-rate effect for dynamic behavior of structures. The conventional limit analysis method has been applied to only static collapse analysis of structures without consideration of dynamic effects in the structural behavior. A dynamic formulation for the limit analysis has been derived for incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ- α method. The dynamic material behavior is described by the Johnson–Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. Simulations have been carried out for impact analysis of a Taylor bar and an S-rail and their numerical results are compared with elasto-plastic explicit analysis results by LS-DYNA3D. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial code compared. The crashworthiness of a structure with the rate-dependent constitutive model is also compared to that with the quasi-static constitutive relation in order to investigate the dynamic effect on deformation of structures.

Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

AbstractThe formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second‐order cone programming (SOCP), for which efficient primal‐dual interior‐point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.

Three-dimensional limit analysis of rigid blocks assemblages. Part I: Torsion failure on frictional interfaces and limit analysis formulation

The governing equations of plastic torsion of, arbitrarily shaped, frictional interfaces are established and a general solution is proposed. The case of rectangular interfaces is used to study the interactions of torsion strength with bending moments and shear forces. In order to solve limit analysis problems, a piecewise linear approximation of the yield function for rectangular interfaces is proposed. A model for the limit analysis of three-dimensional block assemblages interacting through no-tension frictional interfaces is presented including the proposal for the torsion failure mode. This model takes into account non-associated flow rules and limited compressive stresses at the interfaces.

A new discontinuous upper bound limit analysis formulation

AbstractA new upper bound formulation of limit analysis of two‐ and three‐dimensional solids is presented. In contrast to most discrete upper bound methods the present one is formulated in terms of stresses rather than velocities and plastic multipliers. However, by means of duality theory it is shown that the formulation does indeed result in rigorous upper bound solutions. Also, kinematically admissible discontinuities, which have previously been shown to be very efficient, are given an interpretation in terms of stresses. This allows for a much simpler implementation and, in contrast to existing formulations, extension to arbitrary yield criteria in two and three dimensions is straightforward. Finally, the capabilities of the new method are demonstrated through a number of examples. Copyright © 2005 John Wiley & Sons, Ltd.

Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.

Lower bound limit analysis with adaptive remeshing

The objective of this work is to present an adaptive remeshing procedure for lower bound limit analysis with application to soil mechanics. Unlike conventional finite element meshes, a lower bound grid incorporates statically admissible stress discontinuities between adjacent elements. These discontinuities permit large stress jumps over an infinitesimal distance and reduce the number of elements needed to predict the collapse load accurately. In general, the role of the discontinuities is crucial as their arrangement and distribution has a dramatic influence on the accuracy of the lower bound solution (Limit Analysis and Soil Plasticity, 1975). To ensure that the discontinuities are positioned in an optimal manner requires an error estimator and mesh adaptation strategy which accounts for the presence of stress singularities in the computed stress field. Recently, Borges et al. (Int. J. Solids Struct. 2001; 38:1707–1720) presented an anisotropic mesh adaptation strategy for a mixed limit analysis formulation which used a directional error estimator. In the present work, this strategy has been tailored to suit a discontinuous lower bound formulation which employs the stresses and body forces as primary unknowns. The adapted mesh has a maximum density of discontinuities in the direction of the maximum rate of change in the stress field. For problems involving strong stress singularities in the boundary conditions (e.g. a strip footing), the automatic generation of discontinuity fans, centred on the singular points, has been implemented. The efficiency of the proposed technique is demonstrated by analysis of two classical soil mechanics problems; namely the bearing capacity of a rigid strip footing and the collapse of a vertical cut. Copyright © 2005 John Wiley & Sons, Ltd.

A 3D finite element with planar symmetry for limit analysis computations

A formulation for finite element limit analysis of a certain class of 3D perfectly plastic solids governed by von Mises’ plasticity condition is presented. A planar symmetry constraint for both geometry and displacement field is assumed to analyze plane problems where the variable nature of transverse dissipation must be considered. A mixed locking free and low distortion sensitive element is formulated on the basis of the natural approach. The solution procedure exploits the kinematic theorem of limit analysis, cast in the form of a minimum problem for a convex but non-smooth dissipation functional. Applications to a notched specimen and to a bolted joint are presented to stress the importance of transverse effects in some problems commonly modeled as purely 2D.

A directional error estimator for adaptive limit analysis

Limit analysis can be used to directly calculate the ultimate load of structures without considering cumbersome elastic–plastic analysis. We present an adaptive mesh refinement strategy by mean of extended isogeometric analysis (XIGA) in association with second-order cone programming (SOCP) for kinematic limit analysis of hole and inclusion problems. The location of hole and inclusion can be captured by level set function without regard for its interfaces. The local refined non-uniform rational B-splines (LR NURBS), allowing local refinement and modeling complex geometries, are selected as basis functions. The structured mesh refinement strategy is exploited to carry out local refinement on the basis of the indicator of L2-norm of plastic strain rates. The adaptive local refinement can detect the local plastic deformation regions, and further reveals the possible failure mechanisms. The discrete kinematic formulation of limit analysis can be rewritten as the form of SOCP, and further is solved efficiently with the help of the Mosek toolbox. All the desirable features of the present approach are illustrated through numerical experiments.

A mixed finite element model for plane strain limit analysis computations

A formulation for finite element plane strain limit analysis of rigidly perfectly plastic solids governed by von Mises' plasticity condition is presented. The approach is based on the kinematic theorem of limit analysis formulated as a minimum problem for a convex and non-smooth dissipation functional. In particular, the mixed 4-node rectangular element presented by Capsoni and Corradi (1997) is reformulated to keep into account isoparametric mapping while ensuring a locking free behaviour and low distortion sensitivity.