Theorem Of Limit Analysis Research Articles

Lower bound limit analysis of masonry bridges including arch–fill interaction

Collapse tests on full and model scale masonry bridges have shown the structural role of fill and spandrels, which has to be taken into account to obtain realistic evaluations of the load carrying capacity of existing bridges. A plane model of multi-span masonry bridge is proposed in which the vault–fill interaction effects are considered, whose lower bounds on collapse load are obtained by a finite element application of the lower bound theorem of limit analysis. Arches and piers are modelled as beams made up of no tension, ductile in compression material and the fill as a cohesive-frictional material with a tension cut-off. The fill domain is discretized by triangular elements connected by interface elements in order to increase the ratio of the unknowns to the conditions of static admissibility; arches and piers are discretized by two-node straight beam elements. By linearisation of the conditions of plastic admissibility, a Linear Programming problem is formulated and lower bounds on the collapse load are obtained. The procedure is successfully applied to two example bridge models, where a comparison with the results obtained from the kinematic approach is made. The first example is a simulation of a collapse test on a single span bridge; the second concerns a multi-span bridge and highlights the capability of the procedure to describe complex interactions between the arch–pier structural system and the fill at collapse.

Limit analysis of 2-D and 3-D structures based on an ellipsoid yield criterion

In this paper, a nonlinear numerical technique is developed to calculate the limit load and failure mode of structures obeying an ellipsoid yield criterion by means of the kinematic limit theorem, nonlinear programming theory and displacement-based finite element method. Using an associated flow rule, a general yield criterion expressed by an ellipsoid equation can be directly introduced into the kinematic theorem of limit analysis. The yield surface is not linearized and instead a nonlinear purely kinematic formulation is obtained. The nonlinear formulation has a smaller number of constraints and requires less computational effort than a linear formulation. By applying the finite element method, the kinematic limit analysis with an ellipsoid yield criterion is formulated as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the dissipation power which is to be minimized and an upper bound to the plastic limit load of a structure can then be calculated by solving the minimum optimization problem. An effective, direct iterative algorithm has been developed to solve the resulting nonlinear programming formulation. The calculation is based purely on kinematically admissible velocities. The stress field does not need to be calculated and the failure mode of structures can be obtained. The proposed method can be used to calculate the bearing capacity of clay soils in a direct way. Some examples are given to illustrate the validity and effectiveness of the proposed method.

Upper bound limit analysis of active earth pressure with different fracture surface and nonlinear yield criterion

Conventional calculations of static and seismic active earth pressures of soils on a retaining wall are formulated assuming the soils obeying a linear Mohr–Coulomb yield criterion. However, experimental evidences show that the strength envelopes of almost all geomaterials are nonlinear in nature over a wide range of normal stresses. In this paper, the strength envelope of the backfill behind a retaining wall is considered to follow a nonlinear yield criterion. A simple method is proposed for calculating the static and seismic active earth pressures acting against a retaining wall using a nonlinear yield criterion. This method is based on the upper bound theorem of limit analysis. Both translational and rotational fracture surfaces are employed in the formulation for calculating active earth pressures. Quasi-static representation of earthquake effects using a seismic coefficient concept is adopted for seismic active earth pressure calculations. Instead of using directly the actual nonlinear yield criterion, a linear Mohr–Coulomb yield criterion, which is tangential to the nonlinear yield criterion, is used to formulate the active earth pressure problem as a classical nonlinear programming problem. A nonlinear sequential quadratic programming algorithm is used to search for the maximum solution. In order to assess the validity of the proposed method, values of active earth pressures for different values of seismic coefficients and nonlinear parameters in the yield criterion are calculated and compared with solutions obtained using an extended Rankine’s active earth pressure theory. For the case of static active earth pressure, the upper bound solutions using the present method with a translational fracture surface are equal to the extended Rankine’s theoretical solutions and are slightly smaller than those obtained using the present method with a rotational fracture surface. For the case of seismic active earth pressure, numerical results obtained using the present method with a rotational fracture surface is very close to the extended Rankine’s theoretical solutions. A study is conducted to investigate the effects of the parameters in the nonlinear yield criterion on the active earth pressures.

Two-dimensional finite element upper bound limit analysis of masonry bridges

A two-dimensional model for the evaluation of the load carrying capacity of masonry bridges is proposed that takes into account the strengthening effects due to arch–fill interaction observed in experimental tests. Upper bounds on the collapse load and the corresponding mechanism are obtained by means of a finite element application of the Kinematic Theorem of Limit Analysis. Arches and piers are modelled as beams made up of non-tensile resistant (NTR) and ductile in compression masonry, while the fill is represented as a cohesive-frictional material with tension cut-off. Kinematically admissible mechanisms are obtained by discretizing the fill domain with triangular elements and including velocity discontinuities in order to increase the degrees of freedom of the model and thus reduce locking; arches and piers are discretized by two-node straight beam elements. A piecewise linearization of the limit domains allows the upper bound on the collapse load and the corresponding mechanism to be obtained as a solution of a Linear Programming problem. The capabilities and the validity limits of the proposed numerical model are shown in two examples. The collapse test on a real single span bridge is simulated and discussed in the first example; a multi-span bridge is analysed in the second, where complex interactions between arches, piers and fill are obtained.

The ultimate pullout capacity of anchors in frictional soils

During the last 30 years various researchers have proposed approximate techniques to estimate the uplift capacity of soil anchors. As the majority of past research has been experimentally based, much current design practice is based on empiricism. Somewhat surprisingly, very few numerical analyses have been performed to determine the ultimate pullout loads of anchors. This paper presents the results of a rigorous numerical study to estimate the ultimate pullout load for vertical and horizontal plate anchors in frictional soils. Rigorous bounds have been obtained using two numerical procedures that are based on finite element formulations of the upper and lower bound theorems of limit analysis. For comparison purposes, numerical estimates of the break-out factor have also been obtained using the more conventional displacement finite element method. Results are presented in the familiar form of break-out factors based on various soil strength profiles and geometries and are compared with existing numerical and empirical solutions.Key words: anchor, pullout capacity, finite elements, limit analysis, lower bound, sand.

Stress-based upper-bound method and convex optimization: case of the Gurson material

A nonlinear interior point method associated with the kinematic theorem of limit analysis is proposed. Associating these two tools enables one to determine an upper bound of the limit loading of a Gurson material structure from the knowledge of the sole yield criterion. We present the main features of the interior point algorithm and an original method providing a rigorous kinematic bound from a stress formulation of the problem. This method is tested by solving in plane strain the problem of a Gurson infinite bar compressed between rough rigid plates. To cite this article: F. Pastor et al., C. R. Mecanique 334 (2006).

Minimum weight design of solid annular plates under an assigned limit ring load

During the last decades, several authors (Kozlowski W, Mroz Z (1969) Optimal design of solid plates. Int J Solids Struct 5:781–794; Lamblin D, Guerlement G (1976) Dimensionnement plastique de volume minimum sous contraintes de plaques sandwich circulaires soumises a des charges fixes ou mobiles. J Mec 15:55–84; Rozvany GIN, Olhoff N, Cheng KT, Taylor JE (1982) On the solid plate paradox in structural optimization. J Struct Mech 18:1–32) were interested in the minimum weight design of circular and annular plates under a given limit load. The theorems of limit analysis allow to compute the limit load of a given design, and optimality criteria have been established, providing a useful set of tools to solve the problem of weight minimization. However, the application of such criteria can lead to designs of poor practical interest. It is thus interesting to investigate in the field of plates subjected to an additional set of technological constraints and to evaluate the subsequent loss of efficiency in comparison with purely theoretical optima.

Ultimate strength of free-edge composite laminates under tensile loading: A limit analysis approach

An ultimate failure criterion is proposed to predict strength and failure mode of composite laminates under a tensile loading. The upper bound theorem of limit analysis is used to approximate the ultimate load capacity for composite laminates and identify different collapse mechanisms. The present model is used to predict failures modes in plies and interfaces. Experimental results are presented and a comparison with theoretical predictions is made. Tests have been performed on several specimens of various lay-ups. A thickness effect on delamination is observed as well.

Estimation of seismic passive earth pressures with nonlinear failure criterion

Most conventional calculation of seismic passive earth pressures is based on a linear Mohr–Coulomb failure criterion. However, a substantial amount of experimental data shows that almost all geomaterials follow a nonlinear failure criterion in practice. In this paper, the influence of a nonlinear strength envelope is considered in the framework of limit analysis of plasticity. The seismic passive earth pressures on the rigid walls in seismic conditions are estimated using the upper bound theorem of limit analysis in conjunction with the nonlinear failure criterion. A generalized tangential technique, one tangential line to the nonlinear failure criterion, is used to develop the external work and internal energy dissipation. The seismic earth passive pressure is obtained from optimization. An extended Rankine’s theory is proposed with the same nonlinear failure criterion. In order to evaluate the validity of the numerical results using the generalized tangential technique, the solutions of seismic passive earth pressure are also presented in the framework of the extended Rankine’s theory. Numerical results for different seismic coefficients and various nonlinear parameters of the failure criterion are compared with each other. The upper bound solutions using the generalized tangential technique are almost equal to the extended Rankine’s theoretical solutions, with the maximum difference being less than 1%. The good agreement shows that the generalized tangential technique is an effective method for evaluating the seismic passive earth pressure with a nonlinear failure criterion. This paper extends the work of the seismic passive earth pressure using a linear Mohr–Coulomb failure criterion in previous literature to that using the nonlinear failure criterion.

Optimization of plastic spherical shells of von Mises material

An optimization procedure is developed for spherical shells pierced with a central hole. The outer edge of the shell is simply supported whereas the inner edge is absolutely free. The material of the shell is assumed to be an ideal plastic material obeying the von Mises yield condition. Resorting to the lower bound theorem of limit analysis, shells with constant and piece-wise constant thickness are considered. The designs of spherical shells corresponding to maximal load carrying capacity are established for a given weight. Necessary optimality conditions are derived with the aid of variational methods from the theory of optimal control. The obtained set of equations is solved numerically.

Some general results on the stability and flow failure of rigid viscoplastic structures

The evolution of structures made of materials which obey a rigid viscoplastic constitutive law, conceived as a generalisation of the classical Bingham model, and submitted to several loading parameters, is examined here in connection with the limit analysis (or yield design) theorems. A minimum principle for the velocities is established, from which it is shown in particular that the structure remains motionless as far as the loading belongs to the domain of safe loadings, while flow failure is triggered as soon as it is loaded beyond its limit load. Such a property is illustrated on the example of a uniform layer of Bingham material flowing on an inclined plane, due to the combined action of gravity and surface shear loading, considered as two independent loading parameters.

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.

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.

Experimental verification of a (Punch Through Shear) PTS Mode II test for rock

In order to provide more reasonable support design for shallow tunnels in inclined stratum, it is necessary to investigate the collapse of shallow tunnels in inclined stratum. A new collapse mechanism of shallow rectangular and circular tunnels in inclined rock stratum is constructed, which includes four curves. On the basis of the upper bound theorem of limit analysis and the nonlinear Hoek-Brown yield criterion, obtaining the precise expressions of four curves by means of the variational principle, and deriving the equations of the range and total gravity of the potential collapsing block of rectangular and circular tunnels can be solved respectively. The results are consistent with those published and numerical simulation, which confirms the rationality of this research. The parametric sensitivity analysis of the collapse of rectangular and circular tunnels is carried out, which shows that rock mechanics parameters and geometric parameters have a significant impact on the range and total gravity of the potential collapsing block is obvious. Especially, the dip angle of rock stratum plays a key role in the symmetry of tunnel collapse and the progressive collapsing block is unsymmetrical unless the rock strata are horizontal.

A fracture propagation code and its applications to nuclear waste disposal

In order to provide more reasonable support design for shallow tunnels in inclined stratum, it is necessary to investigate the collapse of shallow tunnels in inclined stratum. A new collapse mechanism of shallow rectangular and circular tunnels in inclined rock stratum is constructed, which includes four curves. On the basis of the upper bound theorem of limit analysis and the nonlinear Hoek-Brown yield criterion, obtaining the precise expressions of four curves by means of the variational principle, and deriving the equations of the range and total gravity of the potential collapsing block of rectangular and circular tunnels can be solved respectively. The results are consistent with those published and numerical simulation, which confirms the rationality of this research. The parametric sensitivity analysis of the collapse of rectangular and circular tunnels is carried out, which shows that rock mechanics parameters and geometric parameters have a significant impact on the range and total gravity of the potential collapsing block is obvious. Especially, the dip angle of rock stratum plays a key role in the symmetry of tunnel collapse and the progressive collapsing block is unsymmetrical unless the rock strata are horizontal.

Analysis of Wedge Stability Using Different Methods

The stability problem of a rock slope containing a wedge resting on two intersecting discontinuities is of great interest in rock slope engineering. It is a statistically indeterminate problem with two extra unknowns according to the force (stress) equilibrium analysis. The widely used limit equilibrium methods in practice assume that the directions of the shear forces acting on the two discontinuities are parallel to their line of intersection. The validity of this assumption, however, has not been verified theoretically. This paper presents a general limit equilibrium method that determines the directions of the shear forces by using Pan’s “Maximum principle” and an upper bound method that applies the classic upper bound theorem of limit analysis to avoid making extra assumptions. The formulations of the two methods are derived. A non-symmetric wedge and a symmetric wedge are analyzed using the two derived methods. To further explore the influence on stability due to the direction of the shear force acting on the two discontinuities, three-dimensional finite-element analyses are also conducted. The results are compared and discussed.

Numerical determination of limit loads for three-dimensional structures using boundary element method

Based on the static limit analysis theorem, a solution procedure for limit analysis of three-dimensional (3-D) structures is established making use of conventional boundary element method (BEM) rather than of finite element method (FEM). The self-equilibrium stress fields are expressed by linear combination of several basic self-equilibrium stress fields with parameters to be determined. These basic self-equilibrium stress fields are elastic responses of the body to imposed permanent strains obtained through elastic-plastic incremental analysis by three-dimensional boundary element method (3-D BEM). The Complex method is used to directly solve the resulting nonlinear programming problem and to determine the maximal load amplifier. The numerical results show that it is efficient and accurate to solve 3-D limit analysis problems by using BEM and the Complex method.

Lower bound limit analysis of three-dimensional elastoplastic structures by boundary element method

Based on the lower bound theorem of limit analysis, a solution procedure for limit analysis of three-dimensional elastoplastic structures was established using conventional boundary element method (BEM). The elastic stress field for lower bound limit analysis was computed directly by three-dimensional boundary element method (3-D BEM). The self-equilibrium stress field was constructed by the linear combination of several self-equilibrium “basis vectors” which can be computed by elastic-plastic incremental iteration of 3-D BEM analysis. The lower bound limit analysis problem was finally reduced to a series of nonlinear programming sub-problems with relatively few optimal variables. The complex method was used to solve the nonlinear programming sub-problems. The numerical results show that the present solution procedure has good accuracy and high efficiency.

RC two-way slabs strengthened with CFRP strips: experimental study and a limit analysis approach

This paper deals with strengthening of reinforced concrete (RC) two-way slabs with carbon fibre reinforced plastic (CFRP) strips bonded to the tensile face. The first part deals with an experimental study. The fibre reinforced plastics (FRP) strengthened slab test presents a failure mode with debonding of the external FRP strips from the slab. The second part deals with a limit analysis modelling. The strengthened slab is designed as a three-layered plate. A simplified laminated plate model is used to describe the behaviour of three-layered plate supported in four sides, which is subjected to a load in the centre. The upper bound theorem of limit analysis is used to approximate the ultimate load capacity and identify the different collapse mechanisms. Experimental results are compared with theoretical predictions.

Three-Dimensional Lower Bound Solutions for Stability of Plate Anchors in Clay

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structures include transmission towers, sheet pile walls, and buried pipelines. Although anchors are typically complex in shape (e.g., drag or helical anchors), many previous analyses idealize the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages and the problem can be solved in two dimensions. In contrast to recent numerical studies, this paper applies three-dimensional numerical limit analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in undrained clay. The anchor is idealized as either square, circular, or rectangular in shape. Estimates of the ultimate pullout load are obtained by using a newly developed three-dimensional numerical procedure based on a finite-element formulation of the lower bound theorem of limit analysis. This formulation assumes a perfectly plastic soil model with a Tresca yield criterion. Results are presented in the familiar form of break-out factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions.