A kinematic shakedown theorem considering external loading and temperature variation

A kinematic shakedown theorem including the external loading and temperature variation is derived based on the concept of an assumed yield surface. An example is given to show the efficiency of the theorem obtained.The Project Supported by National Natural Science Foundation of China.     

On kinematic method in shakedown theory: II. Modified kinematic method

A kinematic method for the safety factor calculation is studied in the case of shakedown problem with the polyhedron set of variable loads. The time variable is eliminated, and a simple explicit formula is derived for the kinematic functional when it is considered on a set of regular fields. Conditions are established under which the infimum of functional with respect to such fields equals the safety factor. A finite element discretization of the kinematic extremum problem is considered, and convergence of minimums of the discretized problems to the safety factor is proved.     

Dynamic shakedown by modal analysis

Summary Dynamic shakedown of discrete elastic-perfectly plastic structures under a specified load history is studied using the dynamic characteristics of the structure provided by modal analysis. Several statical and kinematical theorems are presented, including lower and upper bound theorems for the minimum adaptation time of the structure. In the formulation of the kinematical theorems a crucial role is played by the appropriate definition of admissible plastic strain cycle.

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Sommario Si studia il problema dell'adattamento dinamico (shakedown) di una struttura discreta elasto-perfetta-mente plastica e soggetta ad una storia di carichi prestabilita, facendo uso a tale scopo delle caratteristiche dinamiche della struttura fornite dalla analisi modale. Vengono presentati svariati teoremi, sia di tipo statico che cinematico, tra cui taluni teoremi di delimitazione superiore ed inferiore del tempo minimo di adattamento. Nella formulazione dei teoremi cinematici ha un ruolo cruciale la corretta definizione di ciclo deformativo ammissibile.

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This paper is part of a research project sponsored by the National (Italian) Research Council, C.N.R., Group of Structural Engineering, and by the National Department of Education (M.P.I.).     

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.     

A note on formulations of static shakedown analysis with bounded kinematic hardening

Two different formulations for the two-surface model of bounded kinematic hardening can be found in literature on shakedown analysis with the von Mises yield criterion. This short paper explains that these two formulations are not equivalent, although there exists literature asserting that they are equivalent. More specifically, the formulation using the constraint on the stress is not a sufficient condition for the two-surface model. Consequently, the static shakedown analysis using this formulation over-evaluates the shakedown factor in general.     

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.     

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel's inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.     

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.     

Dynamic shakedown of elastic-plastic solids for a set of alternative loading histories

The paper deals with dynamic shakedown of an elastic-perfectly plastic solid body subjected to a loading history which is unknown but is allowed to belong to a given set of loading histories. In the hypothesis of a piecewise linear convex set, a sufficient shakedown theorem is given and a bounding principle for the plastic work produced is formulated in terms of the dynamic elastic responses to a discrete set of loading histories. The solution of a minimization problem gives the most stringent bound which also proves to possess a local character, i.e., it regards the plastic work density at any point.     

Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures

Shakedown analysis is an extension of plastic limit analysis to the case of variable repeated loads and plays a significant role in safety assessment and structural design. This paper presents a solution procedure based on the meshless local Petrov–Galerkin (MLPG) method for lower-bound shakedown analysis of bounded kinematic hardening structures. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the targeted domain. Moreover, the natural neighbour interpolation (NNI) is employed to construct trial functions for simplifying the imposition of essential boundary conditions. The kinematic hardening behaviour is simulated by an overlay model and the numerical difficulties caused by the time parameter are overcome by introducing the conception of load corner. The reduced-basis technique is applied to solve the mathematical programming iteratively through a sequence of reduced residual stress subspaces with very low dimensions and the resulting non-linear programming sub-problems are solved via the Complex method. Numerical examples demonstrate that the proposed solution procedure is feasible and effective to determine the shakedown loads of bounded kinematic hardening structures as well as unbounded kinematic hardening structures.     

A modified basis reduction method for limited kinematic hardening shakedown analysis under complex loads

A novel extension of the basis reduction method for kinematic hardening shakedown problem is presented. Firstly, the basis reduction method is implemented based on the modified Newton–Raphson (N-R) method. Then a new technique for the construction of back stress field is introduced, where the simultaneous influence of multiple load corners in shakedown is taken into consideration. Finally, two typical numerical examples are investigated. The results compared with previous works in literatures demonstrated that the proposed method is accurate and the performance in reducing of computation time is significant.     

Dynamic shakedown theory allowing for second order geometric effects

Summary Two necessary and sufficient conditions for shakedown in a structure subjected to given histories of external loads and imposed strains, in the presence of significant inertia and damping effects, are established. Structural discretization and piecewise linearization of constitutive laws are adopted. This permits to consider several hardening materials and geometric effects, in the spirit of second order theory. The statements can be regarded as a generalization to a broader context of the classical Bleich-Melan and Koiter shakedown theorems. With respect to previous work, the main novelty is the simultaneous coverage of dynamic and second order effects.

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Sommario Si dimostrano due condizioni necessarie e sufficienti per l'adattamento in campo elastico di strutture soggette ad assegnate storie di carichi e distorsioni, in presenza di non trascurabili forze d'inerzia e di smorzamento. La discretizzazione della struttura e la linearizzazione a tratti del legame costitutivo consentono di considerare una larga classe di materiali incrudenti e gli effetti geometrici, nello spirito della teoria del second'ordine. Gli enunciati costituiscono una generalizzazione dei classici teoremi di Bleich-Melan e Koiter. A differenza di risultati precedenti, gli enunciati sono validi in presenza di concomitanti effetti dinamici e geometrici del second'ordine.

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Research sponsored by the (Italian) National Research Committee (CNR).     

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     

Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations

Elastic shakedown for discrete, or finite-element discretized, structures subjected to combinations of static and time-variable loads is addressed in the hypothesis of elastic-perfectly plastic material behavior. The static load is conceived as the weight of an additional mass appended to the structure, whereas the time-variable load is conceived as an unknown sequence of excitations belonging to a specified domain, with intervals between subsequent excitations during which the structure is considered as being motionless. It is shown that, in the plane of the static and time-variable load parameters, the structure's dynamic shakedown domain is nonconvex and that its boundary curve generally exhibits local minima and maxima at those static load values at which a resonant and anti-resonant structural behavior, respectively occurs. It is also shown that, for static loads close to the resonant behavior values, the shakedown limit load can be sensibly smaller than the value computed without taking the appended mass into account. The problem of evaluating the shakedown limit load is discussed and a numerical example presented.     

Dynamic shakedown theory of elastoplastic work-hardening structures allowing for second-order geometric effects

This paper is based on piecewise linear yield surface and discretization of structure. By allowing for inertial force, damping force and second-order geometric effects, the two generalized dynamic shakedown theorems are given for shakedown analysis of structure.The project is supported by National Natural Science Foundation of China and Jilin Provincial Applying-Basic Research Projects.     

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.     

A minimum theorem for cyclic load in excess of shakedown,with application to the evaluation of a ratchet limit

Although the shakedown theorems for perfect plasticity have been known since Koiter's 1960 review paper, extensions of the theory to situations where ratchetting or reverse plasticity occurs in excess of shakedown have not appeared in the literature. In this paper a generalisation of the upper bound theorem is derived which reduces to the upper bound shakedown theorem in the limiting case when the load point approaches the shakedown boundary. The new theory is used to develop a method for identifying the ratchet limit for a class of loading histories through the sequential minimisation of two functionals. A programming method, based on the Elastic Compensation method for shakedown is then derived and convergence proven. Numerical examples of the application of the method to practical problems are discussed by us in an accompanying paper.