Plastic collapse of a circular plate under cyclic loads

The shakedown kinematic approach is used to analyze the collapse modes for a circular plate subjected to cyclic loads distributed axisymmetrically. Depending upon the loading scheme and parameters, the plate may fail by instantaneous, incremental, or alternating plasticity collapse. For quasistatic processes the nonshakedown load may be smaller than the instantaneous plastic limit load. In the case of periodic dynamic loading, which lies outside the framework of plastic limit analysis, the frequency parameter, besides the load amplitudes, can significantly influence the nonshakedown behavior of the plate.     

Failure of a circular reinforced concrete plate against dynamic fluctuating loads

Summary The kinematic shakedown method is used to evaluate the dynamic cyclic collapse loads for a clamped reinforced concrete plate. The method, based on reduced kinematic formulae recently derived by the author, requires determination of the fictitious elastic fields within the plates, in particular – their envelope corresponding to the external loads, and construction of potential incremental collapse mechanisms. The load amplitude-frequency diagrams constructed should help to choose the necessary reinforcements for the plate under given loading conditions. Received 14 January 1998; accepted for publication 25 June 1998     

Shakedown reduced kinematic formulation,separated collapse modes,and numerical implementation

Our shakedown reduced kinematic formulation is developed to solve some typical plane stress problems, using finite element method. Whenever the comparisons are available, our results agree with the available ones in the literature. The advantage of our approach is its simplicity, computational effectiveness, and the separation of collapse modes for possible different treatments. Second-order cone programming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode. The approach is ready to be used to solve general shakedown problems, including those for elastic–plastic kinematic hardening materials and under dynamic loading.     

Dynamic non-shakedown theorem for elastic perfectly-plastic continua

Sufficient and necessary conditions of a kinematic nature are established for alternating plasticity or incremental collapse (i.e. inadaptation or non-shakedown) of elastic perfectly-plastic media subjected to given histories of loads and thermal strains, in the presence of significant inertia and viscous damping forces. The classical second shakedown theorem, due to W.T. Koiter, is thus extended to the dynamic range.     

A bipotential approach for plastic limit loads of strip footings with non-associated materials

This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.     

Shakedown kinematic theorem for elastic–perfectly plastic bodies

The classical shakedown kinematic theorem due to Koiter for elastic–perfectly plastic bodies is re-examined and divided into separated shakedown and nonshakedown theorems. While the shakedown theorem is based on the set of Koiter's plastic strain rate cycles, the non-shakedown one involves a broader set of admissible plastic strain rate cycles, the end-cycle accumulated strains of which are deviatoric parts of compatible strain fields. For certain broad classes of practical problems the two statements are unified to yield the unique theorem in Koiter's sense.     

Failure mechanism for longitudinally stiffened I girders subjected to patch loading

Investigation of the behaviour of plate girders (welded I girders) under the action of localized (patch) loading on a flange in the plane of the web is presented. Girders without vertical (transversal) stiffeners in the zone of load introduction, but with horizontal (longitudinal) stiffeners, are considered. An approximate solution for obtaining ultimate load is developed using failure (collapse) mechanism and following the upper bound theorem of plastic collapse. Procedure is based on the solution introduced by Roberts and Rocky and further elaborated by Roberts for the determination of the ultimate patch load for girders without longitudinal stiffeners. Procedure presented in this paper is adapted for the girders with longitudinal stiffeners. Proposed collapse mechanism is based on the experimental evidence. When the longitudinal stiffeners have enough rigidity, deformation at collapse, buckling of the web, is limited to the part of the web between loaded flange and longitudinal stiffener. Values of the ultimate loads obtained through developed procedure as well as the values from other, earlier authors’ proposal that was accepted and applied in the British standard BS 5400 are compared with the values from authors’ experiments and available database with the test results from other investigators.     

PLASTIC LIMIT LOAD ANALYSIS OF DEFECTIVE PIPELINES

The integrity assessment of defective pipelines represents a practically important task of structural analysis and design in various technological areas, such as oil and gas industry, power plant engineering and chemical factories. An iterative algorithm is presented for the kinematic limit analysis of 3-D rigid-perfectly plastic bodies. A numerical path scheme for radial loading is adopted to deal with complex multi-loading systems. The numerical procedure has been applied to carry out the plastic collapse analysis of pipelines with part-through slot under internal pressure, bending moment and axial force. The effects of various shapes and sizes of part-through slots on the collapse loads of pipelines are systematically investigated and evaluated. Some typical failure modes corresponding to different configurations of slots and loading forms are studied.     

Reduction of Kinematic: Inequality and Optimization of Perfectly Plastic Frameworks

Abstract

In the kinematic theory of structures consisting of perfectly plastic elements, an inequality between the plastic dissipation work and the load work is used. This inequality, which we will term “the kinematic inequality,” must hold for all kinematically admissible mechanisms. These mechanisms are generated by certain parameters which usually remain in the kinematic inequality and which thereby preclude the general application of the kinematic approach. In this paper we overcome this difficulty in the case of frames and provide various applications of the method. By using new theorems we eliminate the parameters and reduce the kinematic inequality to a finite system of inequalities which depend only on frame geometry and on loads. Based on these theorems, a procedure is offered for deriving a system of independent inequalities for general multistory multibay frames. New theorems are then obtained regarding the existence and the rotation of certain plastic hinges in collapse mechanisms. The overall theory is illustrated by a specific example. Finally, the formulations obtained following our method are used to minimize the mass of a fixed-base rectangular portal frame for any length, height, and system of loads.     

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 fast and general upper-bound limit analysis approach for out-of-plane loaded masonry walls

A new general approach for the limit analysis of out-of-plane loaded masonry walls based on an upper bound formulation is presented. A given masonry wall of generic form presenting openings of arbitrary shape is described through its Non-Uniform Rational B-Spline (NURBS) representation in the three-dimensional Euclidean space. A lattice of nodes is defined in the parameters space together with possible fracture lines. An initial set of rigid elements initially subdividing the original wall geometry is identified accordingly. A homogenized upper bound limit analysis formulation, which takes into account the main characteristics of masonry material such as very low resistance in traction and anisotropic behavior is deduced. Moreover the effect of vertical loads and membrane stresses is considered, assuming internal dissipation allowed exclusively along element edges. A number of technically meaningful examples prove that a good estimate of the collapse load multiplier is obtained, provided that the initial net of yield lines is suitably 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.     

A unified expression of elastic–plastic notch stress–strain calculation in bodies subjected to multiaxial cyclic loading

In this paper, a simplified thermodynamics analysis of cyclic plastic deformation is performed in order to establish an energy transition relation for describing the elastic–plastic stress and strain behavior of the notch-tip material element in bodies subjected to multiaxial cyclic loads. Based on the thermodynamics analysis, it is deduced that in the case of elastic–plastic deformation, Neuber’s rule inevitably overestimates the actual stress and strain at the notch tip, while the equivalent strain energy density (ESED) method tends to underestimate the actual notch-tip stress and strain. According to the actual energy conversion occurring in the notch-tip material element during cyclic plastic deformation, a unified expression for estimating the elastic–plastic notch stress–strain responses in bodies subjected to multiaxial cyclic loads is developed, of which Neuber’s rule and the ESED method become two particular cases, i.e. upper and lower bound limits of the notch stress and strain estimations. This expression is verified experimentally under both proportional and non-proportional multiaxial cyclic loads and a good agreement between the calculated and the measured notch strains has been achieved. It is also shown that in the case of multiaxial cyclic loading, the unified expression distinctly improves the accuracy of the notch-tip stress–strain estimations in comparison with Neuber’s rule and the ESED method. The unified expression of the notch stress–strain calculation developed in this paper can thus provide a more logical approximate approach for estimating the elastic–plastic notch-tip stress and strain responses of components subjected to lengthy multiaxial cyclic loading histories for local strain approach-based fatigue-crack-initiation life prediction.     

A kinematic elastic nonshakedown theorem for implicit standard materials

Summary  Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxcé's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated. Received 15 February 2001; accepted for publication 18 October 2001     

A modified upper bound approach to limit analysis for plane strain deeply cracked specimens

The classical upper bound approach of limit analysis is based on assumption of rigid blocks of deformation that move between lines of tangential displacement discontinuity. This assumption leads to considerable simplification but often at cost of higher estimate of the actual load. Moreover, in many cases, it does not give a correct shape of the plastic field. In order to overcome these limitations a modified upper bound approach is proposed in this article. The proposed approach is basically an energetic approach but unlike the classical upper bound approach it is capable of including presence of statically governed stress field. As an application, of proposed approach, theoretical plane strain solutions are presented for deeply cracked fracture mechanics specimens (single edge cracked specimen in pure bending – SE (PB), single edge cracked specimen in three-point bending – SE (B), and compact tension – C (T) specimens). Plane strain plasticity problem in rigid elastic–plastic mono-material (homogeneous) was solved to evaluate useful parameters like limit load, plastic eta function (η_p) and plastic rotation factor (r_p) and in bi-material (mismatch welds) to evaluate mismatch limit load, for deeply cracked specimens. New kinematically admissible velocity fields are proposed for SE (B) and C (T) specimens. Proposed theoretical solutions were confirmed by classical slip-line field solutions, wherever available, and by detailed elastic–plastic finite element analysis with Von-Mises yield criterion. Good agreement was found between proposed solutions and results obtained from the classical slip-line field theory and finite element analysis.     

在冲击载荷作用下弹塑性圆板的反直观动力行为数值分析

对周边简支理想弹塑性圆板受脉冲载荷作用时的动力行为进行了数值计算与分析,揭示了板类结构反直观动力行为的客观存在性．通过分析发现,随着脉冲强度的增加,存在几个窄的载荷区域,板的响应是反直观的,而且在此附近,结构参数、载荷等因素的微小改变将导致响应模式的很大差异,表明反直观行为对这些参数的极其敏感性．进一步计算表明,这一特殊的动力行为主要与板内力间的相互耦合作用密切相关,同时,卸载后的结构反弹到另一侧时发生较大的反向塑性变形,导致能量的进一步耗散,使板呈现反常的动力响应．这一现象是几何与材料两种非线性相互作用的结果。     

On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.     

A kinematic FE limit analysis model for thick English bond masonry walls

Two-wythes masonry walls arranged in English bond texture were often used in the past as bearing panels in seismic area. On the other hand, earthquake surveys have demonstrated that masonry strength under horizontal actions is usually insufficient, causing premature collapses of masonry buildings, often ascribed to out-of-plane actions. Furthermore, many codes of practice impose for new brickwork walls a minimal slenderness, which for instance is fixed by the Italian O.P.C.M. 3431 equal to 12 for artificial bricks and 10 for natural blocks masonry.For the above reasons, the analysis at failure of English bond brickwork walls under out-of-plane actions is a topic that deserves consideration, despite the fact that almost the totality of the studies of masonry at failure is devoted to running bond arrangements. Furthermore, it must be noted that an approach based on the analysis of running bond texture – in comparison with English bond pattern – is not suitable for the investigation of the behavior of bearing panels.In this framework, in the present paper, a Reissner–Mindlin kinematic limit analysis approach is presented for the derivation of the macroscopic failure surfaces of two-wythes masonry arranged in English bond texture. In particular, the behavior of a 3D system constituted by infinitely resistant bricks connected by joints reduced to interfaces with frictional behavior and limited tensile/compressive strength is identified with a 2D Reissner–Mindlin plate. In this way, assuming both an associated flow rule for the constituent materials and a finite subclass of possible deformation modes, an upper bound approximation of macroscopic English bond masonry failure surfaces is obtained as a function of macroscopic bending moments, torsion and shear forces.Several examples of technical relevance are treated both at a cell level and at a structural level, addressing the differences in terms of collapse loads and failure surfaces due to different textures and constituent laws for joints. Finally, two meaningful structural examples consisting of a panel in cylindrical flexion and a masonry slab constrained at three edges and out-of-plane loaded are discussed. A detailed comparison in terms of deformed shapes at collapse and failure loads between a 2D FE Reissner–Mindlin limit analysis approach and a full 3D heterogeneous FE model shows the reliability of the results obtained using the kinematic identification approach proposed.     

A plasticity model for cellular materials with open-celled structure

Based on a rigid-plastic material model that obeys the von Mises yield criterion, the plastic behavior of foams with an open-celled structure is studied in this paper using a single unit cell. An approximate continuum plasticity model is developed within the framework of the upper bound theorem of plasticity to describe the yield behavior of foams. The microscopic velocity fields are derived for the unit cell, which satisfy the incompressibility and the kinematic boundary conditions, and expressed in macroscopic rate of deformation. From the microscopic velocity fields, a macroscopic yield function is developed for foams under multi-axial stresses and includes the effects of the hydrostatic stress due to the void presence and growth. The dependency of the derived yield surfaces of foams on their relative densities is studied. The plastic behavior of foams is also studied numerically using the finite element method. The newly developed plasticity model is compared with the finite element analysis results and other available foam models and then correlated with the finite element results.     

Application of the kinematical shakedown theorem to rolling and sliding point contacts

In the plane-strain conditions of a long cylinder in rolling line contact with an elastic-perfectly-plastic half-space an exact shakedown limit has been established previously by use of both the statical (lower bound) and kinematical (upper bound) shakedown theorems. At loads above this limit incremental strain growth or “ratchetting” takes place by a mechanism in which surface layers are plastically sheared relative to the subsurface material.In this paper the kinematical shakedown theorem is used to investigate this mode of deformation for rolling and sliding point contacts, in which a Hertz pressure and frictional traction act on an elliptical area which repeatedly traverses the surface of a half-space. Although a similar mechanism of incremental collapse is possible, the behaviour is found to be different from that in two-dimensional line contact in three significant ways: (i) To develop a mechanism for incremental growth the plastic shear zone must spread to the surface at the sides of the contact so that a complete segment of material immediately beneath the loaded area is free to displace relative to the remainder of the half-space, (ii) Residual shear stresses orthogonal to the surface are developed in the subsurface layers, (iii) A range of loads is found in which a closed cycle of alternating plasticity takes place without incremental growth, a condition often referred to as “plastic shakedown”.Optimal upper bounds to both the elastic and plastic shakedown limits have been found for varying coefficients of traction and shapes of the loaded ellipse. The analysis also gives estimates of the residual orthogonal shear stresses which are induced.     

Energy absorption of an egg-box material

Conical frustra made from leaded gun-metal have been compressed axially. Collapse is either by a travelling plastic hinge or by tearing. An analytical model is developed for the travelling plastic hinge in a rigid, ideally plastic solid; its predictions are compared with the observed response, and with those of an axisymmetric finite element analysis. The travelling hinge mechanism is also observed in the compressive collapse of an egg-box material comprising a square array of conical frustra. Collapse mechanism maps are constructed for the egg-box material, and they show the regimes of dominance of elastic buckling, material tearing and the travelling plastic hinge. The maps are useful for selecting egg-box geometries that maximise the energy absorption per unit mass at any prescribed value of collapse stress. The optimisation indicates that the egg-box material has a similar energy absorption capacity to that of hexagonal honeycombs and is superior to that of metal foams.