Optimal shakedown loading for circular plates

Optimization of shakedown loading under constrained residual displacement is considered for elastic and perfectly plastic circular plates. The load variation bounds, which satisfy the optimality criterion in concert with plate-strength and stiffness requirements, are identified. The actual strain fields of the plate depend on the loading history. Thus, the load optimization problem at shakedown is stated as a pair of problems that are executed in parallel: the main load optimization and the verification of the prescribed magnitudes of the bounds on the residual deflections. The problem must be solved by iteration. The Rozen projected gradient method is applied. A mechanical interpretation of the Rozen optimality criterion is given, which permits the simplification of the mathematical model for load optimization and the formulation of the solution algorithm. Numerical examples include circular annular plates with and without a rigid inclusion. The results are valid under the assumption of small displacements.     

Adaptation of rigid-work-hardening discrete structures subjected to load and temperature cycles and second-order geometric effects

W. Prager has recently introduced the notion of rigid-plastic shakedown. The present paper gives: (1) an extension of Prager's shakedown theorem to the general case of discrete structures and for a broader class of hardening rules, with temperature and geometric effects included; (2) two different methods of bounding maximum shakedown deflections.The results obtained turn out to be a nontrivial analogy of the elastoplastic shakedown theory. Essential differences occur because there is no direct relationship between plastic deformations and residual stresses in the case of rigid-plastic structures.     

Evaluating shakedown under proportional loading by non-linear static analysis

A lower bound method for calculating shakedown loads under proportional loading by static non-linear finite element analysis is presented. Stress fields obtained by static analysis and stress superposition are substituted into Melan’s lower bound shakedown theorem. The proposed method is applied to two sample problems: a thick cylinder under internal pressure and a square plate with a central hole under proportional biaxial loading. The results indicate that the method gives accurate lower bound shakedown loads for these problems.     

Solution validation technique for optimal shakedown design problems

Modern computer technology allows conjoining shakedown theory, optimization and ever stricter standardized design requirements in a single mathematical problem formulation. However it raises a question of reliability: easily achieved solution should not be taken for granted but should be adequately assessed. This paper focuses on the physical validation technique for optimal shakedown design problem solution in the aspect of Melan theorem (statics) and residual deformation compatibility (kinematics). For that purpose Rosen gradient projection method is used. Optimization problem of bending circular, symmetric plate at shakedown, which is subjected by a variable repeated load, is considered for illustration of the validation technique.     

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.     

Computational methods for optimal shakedown design of FE structures

The paper concerns the optimal shakedown design of structures discretized by elastic perfectly plastic finite elements. The design problem is formulated in four alternative versions, i.e. as the search for the minimum volume design whose shakedown limit load multiplier is assigned or as the search for the maximum shakedown limit load multiplier design whose volume is assigned; both problems are approached on the grounds of the shakedown lower bound and upper bound theorems. Correspondingly four computational methods, one for each original problem, are presented. These methods consist in solving iteratively new problems which are simpler than the original ones, but expressed in such a way that the obtained design and behavioural variables fulfill the optimality conditions of the relevant original problems, and thus they provide the true optimal design. Finally, an alternative numerical approach devoted to obtaining the optimal shakedown design is presented. Several numerical examples confirm the theoretical results.     

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

A simplified method of solution for the short cycle creep-plasticity problem

This paper describes a method for estimating the long-term effects on structures under cyclic changes of loading. For loads less than a certain critical amplitude (shakedown limit), the stress in the structure will a symptote to a cyclic stationary state consisting of an elastic part in response to the cyclic loading, plus a system of self-equilibrating residuals constant in time. It is shown that corresponding to this cyclic stationary state, the creep energy dissipation per cycle of loading is a maximum. Instead of following the exact time history to reach this state, in this paper it is found by a procedure of successive approximations. It corrects the admissible residual stress distribution at the beginning of a cycle by the creep and plastic strains accumulated over an entire cycle, which are in general not compatible, and requires additional self-equilibrating stresses to give an elastic strain distribution such that the total strain satisfies compatibility. The steady state is reached when no further correction is necessary. Convergence may be accelerated by a suitable choice of initial starting value, and by an artificial choice of the cycle time for the best computational convenience, upon which the steady-state solution can be proved to be independent. The procedure is a powerful device to obtain the cyclic steady-state solution, which will give an upper bound to the creep deformation per cycle and may also be used to find the shakedown limit. The formulation of the procedure in conjunction with the finite element method is given in detail and results of a few examples of the analysis are shown.     

Optimum shakedown design under residual displacement constraints

We consider the minimum weight design of suitably discretized elastoplastic structures subjected to variable repeated loads and constraints on the amount of residual (or permanent) deflections. The optimization problem is formulated on the basis of the classical lower bound theorem of shakedown, supplemented by appropriate constraints on deflections obtained from existing bounding results. The primary purpose of the paper is to show that, even for large size structures, this important and challenging problem can be modeled and solved directly as a mathematical programming problem within the industry standard modeling framework GAMS. Examples concerning truss-like structures are presented for illustrative purposes. Received January 18, 1999     

Computer shakedown analysis of planar structures

The paper discusses computer applications in shakedown analysis of structures. It emphasizes on the need of appropriate and easy to handle softwares for practical engineering uses. The shakedown analysis capability of a general, inelastic computer software is briefly explained and illustrated by an example.     

On a numerical approach to shakedown analysis of structures

Two alternative ways of performing the shakedown analysis of structures subjected to variable repeated loads are presented. The first one enables one to check whether the structure shakes down (i.e. responds elastically after a few elastoplastic cycles) or not. Such a check is done by reproducing incrementally a critical cyclic load which corresponds to a cyclic repetition of piecewise-proportional load path that contains all the vertices of the variable load domain. The second approach enables one to find a safety factor for the limit of shakedown capability. The problem is one of convex or linear programming, depending on the kind of yield condition used. Numerical results presented in the paper show that the general purpose software that performs incremental elastoplastic calculations can be successfully used for shakedown analysis.     

Numerical lower bound shakedown analysis of engineering structures

We propose a numerical method for the computation of shakedown loads of engineering structures subjected to varying thermo-mechanical loading. The method is based on Melan’s lower bound shakedown theorem using the von Mises yield criterion. The resulting nonlinear convex optimization problem is presented in a generalized formulation and then solved by an interior-point algorithm, which is characterized by a problem-tailored solution strategy, particularly suitable for application to large-scale engineering structures.Theoretical and numerical issues of the algorithm are described. It’s efficiency is shown by application to thermo-mechanical problems from power plant engineering. The results are compared to those found in literature as well as to calculations with other optimization codes lancelot, ipopt and ipdca.     

Reliability-based design optimization of trusses under dynamic shakedown constraints

Structural and Multidisciplinary Optimization - A reliability-based design optimization problem under dynamic shakedown constraints for elastic perfectly plastic truss structures subjected to...     

Optimal shakedown design of frames by linear programming

A mathematical model of flexural framed structures subjected to cyclic loadings is presented, and the problem of minimum-weight shakedown design of these structures is formulated as a pair of dual linear programming problems. Such a formulation reflects the duality between static and kinematic theorems of shakedown theory. An optimal solution for plastic moment distribution realizes a minimum volume of a structure and ensures a given safety factor against plastic collapse by inadaptation to a prescribed cyclic loading program. The reduced-size form of the problem is presented where the bending moments are expressed in terms of independent redundants. This form is more attractive from the computational point of view for the use of linear programming codes. Several examples illustrate the presented method.     

Shakedown state simulation techniques based on linear elastic solutions

The paper extends an iterative method, previously applied to the determination of the limit state of a perfectly plastic body [5], to the determination of the shakedown limit state of a body subjected to a cyclic history of mechanical load and temperature. A convergence proof is presented for a particular class of problems where the magnitude of a mechanical load at the shakedown limit is found as the limit of a sequence of monotonically reducing upper bounds. An implementation of the method in a finite element scheme is discussed and examples of the application of the method to sample problems are presented.     

Discrete optimal design of elasto-plastic trusses using compliance and stability constraints

In this paper two discrete optimization methods are presented for the minimum volume design of elasto-plastic trusses with given geometry. The design is based on given sets of discrete cross-sectional sizes. Both methods enable the use of the plastic reserve of the truss; the plastic deformations, however, are controlled by compliance constraints on plastic deformations. In the second solution method, the shakedown of the truss is also taken into consideration. The stability of the bars is also controlled by using permissible stresses for compression. The methods are based on continuous optimal elasto-plastic design methods and on the discrete optimization method of elastic trusses using segmental approach. By the iterative application of these methods, solution procedures that use standard linear programming have been developed.Dedicated to Prof. F. Ziegler, for his 60-th birthday; extended version of a paper presented at the Second World congress of Structural and Multidisciplinary Optimization (WCSMO-2), in Zakopane, Poland, May 1997     

Plane strain sliding contact of multilayer magnetic storage thin-films using the finite element method

The sliding contact or scratch behavior of multi-layer thin-films such as those found in magnetic storage disks has been studied using the finite element method. A rigid cylinder sliding over a multilayered thin-film half-space was implemented to simulate the contact between a feature of the recording slider (such as the protrusion on the trailing edge of the slider, which is part of the thermal flying-height control, TFC) and the magnetic storage multilayer disk. The effects of different parameters such as normal load, friction coefficient and TFC radius on the von Mises, shear and principal stresses in the multilayer system were analyzed. Results showed that under sliding conditions, for a given normal load, the friction coefficient influences the location and magnitude of the plastic strain in the multilayer system. Repeated sliding contact was also performed to characterize its effect on the stress and strain behavior under various loading conditions and investigate shakedown behavior.     

Computational procedures for plastic shakedown design of structures

The minimum volume design problem of elastic perfectly plastic finite element structures subjected to a combination of fixed and perfect cyclic loads is studied. The design problem is formulated in such a way that incremental collapse is certainly prevented. The search for the structural design with the required limit behaviour is effected following two different formulations, both developed on the grounds of a statical approach: the first one operates below the elastic shakedown limit and is able to provide a suboptimal design; the second one operates above the elastic shakedown limit and is able to provide the/an optimal design. The Kuhn–Tucker conditions of the two problems provide useful information about the different behaviour of the obtained structures.An application concludes the paper; the comparison among the designs is effected, pointing out the different behaviour of the obtained structures as well as the required computational effort related to the numerical solutions.     

ERP evaluation during the shakedown phase: lessons from an after-sales division

Abstract.  Integrated information systems and operational efficiency are both pivotal issues for contemporary firms. While there is substantial evidence that enterprise resources planning (ERP) systems can deliver related improvements, the implementation of these systems can turn out to be a very complex and risky task. One key aspect of managing these risks is maintaining operational momentum and preventing possible problems from escalating in the so-called shakedown phase shortly after system implementation. The objective of this paper is to examine how user evaluations of ERP system success could be used to trace down the source of potential problems, which can arise during the shakedown phase, and how and why experienced system success might vary between different user groups. The paper builds on a case study completed in the after-sales division of a large multinational organization. This context is considered a fruitful empirical setting for the study as the business sets enormous demands for ERP system functionality and its smooth implementation. Based on the case study, it is argued that in such a context the importance of sufficient user skills, data reliability and intra-organizational communication becomes emphasized in the ERP implementation process. Moreover, results illustrate how downstream operations and customer relations are particularly vulnerable to problems accumulated in upstream business processes. Related problems can potentially form a self-fulfilling cycle, where the lack of skills and information constantly deteriorates both user perceptions and actual operational performance.     

A linear programming approach to shakedown analysis of structures

Two and three dimensional structures are dealt with, subjected to variable repeated loads, in order to establish a numerical tool for determining the load domain multiplier that gives rise to shakedown. The structure is made discrete by finite elements and the yield domain is linearized. By applying Bleich and Melan's theorem, two primal static formulations are found in linear programming, from which the relevant dual kinematic versions are obtained via duality properties.Numerical results are given at the end of the paper, together with some considerations about the numerical efficiency of the proposed formulations.