Piecewise Linear Yield Surface Research Articles

An improved adaptive formulation for the computation of limit analysis problems of thin axisymmetrical shells

An improved adaptive technique for the finite element formulation of limit analysis problems of symmetrically loaded thin shells of revolution is presented. Bending, membrane behavior and changes of curvature are simulated by using an axisymmetrical thin shell element of three nodes associated with a new piecewise linear (PWL) yield surface. The results obtained show better upper bounds when compared to other numerical and analytical solutions reported in the literature. Finally, an a posteriori error measure is re-written to include the new displacement rate and PWL strain rate fields, resulting in an adaptive scheme for refining the finite element mesh.

A constitutive model for polymers with a piecewise linear yield surface

AbstractReliable prediction of the behaviour of structures made from polymers, especially under high dynamic loading, is a topic under considerable investigation in engineering practice. A constitutive model is derived including important phenomena like necking, crazing and strain rate dependency. In particular, different yield surfaces in compression and tension are taken into account using a piecewise linear formulation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A posteriori error estimator and adaptive procedures for computation of shakedown and limit loads on pressure vessels

Adaptive finite element procedures are presented for the computation of upper bounds estimates of limit and shakedown loads for pressure vessels. The method consists of an h-type adaptive mesh refinement strategy based upon an a-posteriori error estimator measured by the energy norm. The problem is formulated in a kinematic approach using Koiter's shakedown theorem. A constitutive model, for elastic-perfectly plastic materials, relates the plastic strains increments and curvatures to plastic multipliers through the flow law associated with a shell piecewise-linear yield surface (hexagonal prism). A consistent relationship between nodal displacements and nodal plastic multipliers is enforced by minimizing the strain residual between the total strain and the plastic strain increments, which is measured with respect to the energy norm. Discretization of the shell into finite elements allows the reduction of the problem to a minimization problem which is solved by linear programming.

LARGE DISPLACEMENT ELASTOPLASTIC ANALYSIS OF SEMIRIGID STEEL FRAMES

A mathematical programming approach for the large displacement analysis of elastoplastic plane frames with flexible, partial-strength beam-to-column connections is proposed. The incremental formulation is developed within the framework of a discrete model, piecewise linear yield surfaces and large displacement theory. It is based on the fundamental relations of equilibrium, compatibility and constitutive laws, all expressed in incremental form. The irreversible plasticity laws are used in a stepwise holonomic format. A feature of the final mathematical programming problem, known as a non-linear complementarity problem, is that the governing relations exhibit symmetry as a result of the use of fictitious quantities and non-linear residuals. The iterative predictor–corrector computational scheme adopted has the ability of tracing historically a sequence of plastic hinge activations and/or unloadings well beyond any critical point. Numerical examples are presented to illustrate and validate the accuracy of the approach.

LARGE DISPLACEMENT ELASTOPLASTIC ANALYSIS OF SEMIRIGID STEEL FRAMES

A mathematical programming approach for the large displacement analysis of elastoplastic plane frames with flexible, partial-strength beam-to-column connections is proposed. The incremental formulation is developed within the framework of a discrete model, piecewise linear yield surfaces and large displacement theory. It is based on the fundamental relations of equilibrium, compatibility and constitutive laws, all expressed in incremental form. The irreversible plasticity laws are used in a stepwise holonomic format. A feature of the final mathematical programming problem, known as a non-linear complementarity problem, is that the governing relations exhibit symmetry as a result of the use of fictitious quantities and non-linear residuals. The iterative predictor–corrector computational scheme adopted has the ability of tracing historically a sequence of plastic hinge activations and/or unloadings well beyond any critical point. Numerical examples are presented to illustrate and validate the accuracy of the approach.

Shakedown of Frames with Semirigid Connections

The shakedown analysis of elastoplastic plane frames with semirigid connections is considered within a framework of discrete models and piecewise linear yield surfaces. The formulation adopted is based on the classical Bleich‐Melan static theorem, and leads to a linear programming problem. A simple linear elastic perfectly plastic connection element is used to account for the reduced flexibility and partial strength of semirigid beam‐to‐column connections. The necessary conditions of equilibrium and of yield conformity are expressed compactly in matrix form, and the calculation of the elastic locus for the cyclic‐load domain is clarified. A means of identifying whether incremental collapse or alternating plasticity governs failure under repeated loading is also mentioned. Finally, three examples are presented primarily as a preliminary assessment of the shakedown behavior of semirigid frames. In particular, it is shown that the shakedown limit of such structures can be significantly lower than the corresponding plastic collapse limit. The importance of considering axial effects on plasticity and of using the correct elastic stiffness in shakedown calculations are also highlighted.

Extremal formulations for the single finite time step analysis of elastoplastic continua

For the elastoplastic non-holonomic continuum problem two extremal formulations are given for the complete space-time solution of a finite time interval of the loading process. The elastoplastic behaviour is of associated type with piecewise-linear yield surface and linear hardening. In the case of infinitesimal time interval or of finite time interval with holonomic material behaviour the above functional reduces to known formulations for the incremental problem or for the finite step holonomic problem respectively.

Internal variable formulation of a backward difference corrector algorithm for piecewise linear yield surfaces

The formulation of a backward difference algorithm based on an internal variable description for piecewise linear yield surfaces is presented. Attention is restricted to an associated flow rule and isotropic material behaviour. The Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantages of being fully linked to the governing principles and avoiding the inherent problems associated with corners on the yield surface. It is used to identify return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules. These return paths provide a basis against which heuristically developed algorithms can be compared.

Boundary element analysis by linearized nonlinear elastic material models: an application to no-tension systems

This paper deals with the analysis of nonlinear elastic systems discretized by boundary elements in the presence of piecewise-linear material models. The problem is solved in the context of the ‘holonomic’ theory (or ‘deformation’ theory) of plasticity, plane-stress conditions only are dealt with and the implications of linearizing the relevant yield surface is briefly discussed. Piecewise-linear yield surfaces coupled with linear hardening lead to a linear complementarity problem, for which a step-by-step solution procedure is proposed. The paper gives details about the algorithm implemented and discusses its application with reference to the boundary element method. To illustrate the problem with reference to simple but still real systems, masonry walls are considered and a no-tension material model is assumed. Numerical results are presented and compared with those provided by programs that follow different approaches, but use the same assumptions (a no-tension material model with nonlinear, reversible behaviour): namely, a boundary element code based on a conventional ‘initial stress’ procedure and a finite element package.

Indirect identification of yield limits by mathematical programming

An elastic-perfectly plastic discretized structure subjected to given proportional loads, undergoes displacements, some of which are measured; on the basis of these experimental data assumed as exact (unaffected by measurement errors) the yield limits are sought, whereas the elastic and geometric properties are known. This special problem of identification (or inverse problem in structural elasto-plasticity), under suitable hypotheses of piecewise linear yield surfaces and no local unstressing under increasing loads is shown to be amenable to the minimization of a convex quadratic function under linear and (nonconvex) complementarity constraints. Various alternative solution procedures are proposed. Among these the most promising method from the computational standpoint, consists of two phases: (1) minimizing, under linear constraints only, a nonconvex quadratic function of the plastic strains and of the parameters to identify (thus enforcing by penalization the fulfilment of complementarity); (II) checking the optimality of the solution (and moving from a possible local minimum to the global one) by a procedure based on recent results from quadratic complementarity theory. The mathematical programming method developed for the indirect identification of yield limits is tested by means of examples with encouraging results.

On Work-Hardening Adaptation of Discrete Structures Subjected to Dynamic Forces in the Presence of Second-Order Geometric Effects*

ABSTRACT The present paper deals with work-hardening adaptation of discrete rigid-plastic structures subjected to loadings which vary within a given domain according to an unknown history, with inertia forces, viscous forces, and second-order geometric effects also being included. Considering structural elements characterized by a piecewise linear yield surface, a piecewise linear work-hardening law, and a linear strain-rate sensitivity law, we give an adaptation criterion of “statical” type as well as a method of providing a priori bounds to deformation parameters such as displacement, plastic strain, and plastic strain intensity. These bounds can be rendered most stringent by solving a minimization problem of mathematical programming on an equivalent boundary value problem of finite plasticity. A simple application concludes the paper.

Optimum design of rigidworkhardening structures with constraints on deformation

The problem of optimum design of rigid-plastic discrete structures is studied adopting as the design criterion the workhardening adaptation concept, the latter being meant in a narrow sense. According to this, a safe structure is one not only able to adapt to statically or dynamically variable loads inside a given domain, but also strong enough for some deformation parameters (such as displacement, strain and plastic strain intensity components) to be not greater than given upper limits. This control over deformation is made possible by starting two bounding theorems based on a perturbation procedure of the bounding technique of shakedown theory. In spite of some simplifying hypotheses (linear cost function and plastic resistances, piecewise linear yield surface and workhardening law, linear strain-rate sensitivity, etc) the optimization problem is a nonlinear one of mathematical programming. The Kuhn-Tucker conditions are discussed. A simple application is given and future developments are considered.

Bounds to internal forces for elastic-plastic solids subjected to variable loads

Considering an elastic-plastic workhardening solid with piecewise linear yield surfaces and a piecewise linear workhardening law, we give a method for constructing bounds to the internal forces and to the (hardened) yield stresses produced by the action of variable loads at any point of the body and at any time. The loading history is supposed to be unknown, but the loads range within a given domain.

Some Upper Bound Principles to Plastic Strains in Dynamic Shakedown of Elastoplastic Structures

ABSTRACT Suitable measures of plastic strains after adaptation (shakedown) of elastoplastic continuous structures subjected to dynamic loading are shown to be bounded from above by quantities which can be evaluated on the basis of a linear elastic response of the body to prescribed initial conditions. The results achieved are extended to elastic work-hardening bodies under the assumption of a piecewise linear yield surface. The practical use of the results is checked through application to a simple example.

A unified approach to quasi-static shakedown problems for elastic-plastic solids with piecewise linear yield surface

The paper concerns shakedown analysis of elastic-plastic bodies subjected to quasi-statically varying loads within a given domain. Using a perturbation method, a general inequality is given, from which, by simply specializing the perturbing terms, the generalized Melan theorem as well as bounds on various deformation parameters (such as displacements or plastic strain intensities) are derived. The solution of the «perturbed» shakedown problem in finite or holonimic terms permits the bound to be the most stringent and expressible in «local» terms instead of integral terms. A simple application concludes the paper.

Adaptation of Rigid-Plastic Continua under Dynamic Loadings

ABSTRACT The paper deals with the work-hardening adaptation of rigid-plastic solids subjected to arbitrary dynamic loads whose components are within given limits. Sufficient and necessary criteria of “statical” type are presented. The material is characterized by piecewise linear yield surface, piecewise linear work-hardening, and linear strain-rate sensitivity. The displacements are assumed to be infinitesimal.

On plastic dynamic flexure of plates

This paper discusses the rigid-plastic flexure of plates subjected to dynamic loading. A piecewise linear yield surface and small displacements are assumed. The plate is discretized by assuming the constant moment mixed triangular element of Herrmann. Under the above assumptions the dynamic loading of plates is turned into a problem in linear inequalities which is a priori discretized in time. An approximate technique is developed, which takes into account the influence of the strain-hardening and the strain rate sensitivity of material. Some numerical examples of plates subjected to a given velocity distribution confirm the efficiency of this approach.

Local Unloading in Piecewise-Linear Plasticity

Piecewise-linear yield surfaces and hardening rules are assumed for structural constituents. Using results due to Cottle on parametric linear complementarity problems, criteria are given for regular progression (no local unloading) in the plastic deformation response: (1) To proportional paths of generalized stresses and, subsequently, generalized strains at the element level; and (2) to proportional paths of external actions (loads, imposed strains, or both) at the (assembled) structure level.

On plastic dynamics of discrete structural models

This paper discusses the dynamic loading of rigid-perfectly plastic structures by adopting a structural model discretized with constant stress finite elements. This assumption together with the hypothesis of small displacements and of a piecewise linear yield surface leads to the formulation of a problem in linear inequalities, which is, implicitly, time discretized. It is shown how the non linearity of the associated plastic flow laws leads to an equivalent extremal formulation. A numerical algorithm for solving the problem is directly derived from the mechanical statements. Numerical examples illustrate this approach. In the appendix an approximate technique is developed, which takes into account the influence of the strain-hardening and the strain rate sensitivity of material.

Extremum theorems for large displacement analysis of discrete elastoplastic structures with piecewise linear yield surfaces

Discrete or discretized structures are considered in the range of large displacements. Elastic plastic behavior is assumed, under the hypothesis that both yield functions and hardening rules are piecewise linear. The structural response to a single finite loading step is assumed to involve regularly progressive yielding (no local unloading). An extremum property of this structural response is established, by recognizing that the relations governing the configuration change coincide with the Kuhn-Tucker conditions of a particular nonlinear constrained optimization problem, subject to sign constraints alone. This extremum property can be regarded as an extension of the theorem of minimum potential energy. Other properties, even if computationally less attractive, broaden the theory developed, so that some results previously obtained are derived as special cases.