Shakedown Theorem Research Articles

A linear programming upper bound approach to the shakedown limit of thin shells subjected to variable thermal loading

This paper describes an upper bound technique for the evaluation of the shakedown limit of thin cyclindrical shells subject to thermal loading. The method is based upon the upper bound kinematic shakedown theorem of Koiter. By suitable choice of displacement field, in a finite element form, and yield surface, the problem is reduced to a linear programming problem. A number of solutions are presented involving a tube subjected to a moving temperature front which indicates that the technique provides, in an efficient way, a complete description of the load levels at which ratchetting would occur and the corresponding modes of deformation. The technique seems therefore, to provide a useful intermediary between the use of the simple rules incorporated in design codes and full inelastic analysis.

The influence of transient thermal loading on the Bree plate; A simplified method of analysis

The paper describes a simplified technique, based upon an extension of the upper bound shakedown theorem, for the evaluation of ratchet boundaries for a plate subject to a through-thickness temperature transient and in-plane loading. The study of a range of cases indicates that transient effects can have a significant effect upon the deformation properties and that strain growth can occur for zero applied loads at quite moderate levels of thermal loading. As a result the identification of transient thermal stresses as F stresses in ASME design codes does not seem acceptable.

Bemessungsvorschlag für kegelförmige Böden unter innerem Überdruck

A new proposal for the design of conical and toriconical pressure vessel ends is presented. Using the results obtained from a numerically computed elastic stress analysis for a series of geometrical parameters, approximation functions for stress concentration factors have been found. Charts for determining the wall thickness were obtained by applying an approach based on the shakedown theorem as a design criterion. The procedure is intended to be a basis for the establishment of design rules for conical pressure vessel ends in accordance with the present state of the art. The approximation functions derived facilitate the programming of this method.

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.

Interaction buckling-progressive deformation

The interaction between buckling and ratchetting under cyclic additional loading is discussed for elastic-plastic structures. It is shown that progressive plastic deformation can lead to the buckling of a structure as a consequence of the resulting additional displacement. This interaction is particularly strong in perfect plasticity. In the case of kinematic hardening materials, an estimate of the primary loading is given in order to prevent the risk of ratchetting. It is shown that elastic and plastic shake-down theorems still hold when the primary load is smaller than the smallest critical load of tangent modulus.

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.

Dynamic shakedown theory allowing for second order geometric effects

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.

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.

Inadaptation theorems in the dynamics of elastic-work hardening structures

Some broad classes of discrete structural models and piecewise linear yield loci and hardening rules are considered. The dynamic elastoplastic response to a given history of rapidly variable loads and imposed strains and displacements (“dislocations”) is studied on the basis of a suitable matrix description. Inadaptation means that the plastic work and, hence, some plastic deformations increase unlimitedly in time, i.e. the structure does not shakedown. Sufficient and necessary conditions for this occurrence are established and formulated in a theorem, which represents the extension to the dynamic range and to work-hardening structures of the second (Koiter's) shakedown theorem of classical plasticity.

Shakedown of Pressure Vessels with Ellipsoidal Heads

The shakedown of cylindrical pressure vessels with arbitrary heads is described in general by means of Leckie's adaptation of Melan's shakedown theorem. The procedure is mated with an elastic shell analysis programme, which is then used to calculate shakedown pressures for vessels with ellipsoidal heads that have aspect ratios of 3, 2 1/2, 2, 1 1/2, 1, and a variety of thickness ratios. Comparisons with the design pressures obtained by the methods of the A.S.M.E. Boiler and Pressure Vessel Code are also made.

Shakedown in Elastic-Plastic Systems Under Dynamic Loadings

Following the derivations of Koiter, who gave a general proof of Melan’s shakedown theorem for elastic-plastic systems under quasi-static reversible parametric loadings, it is shown here that the inclusion of the inertia force due to dynamic loadings does not change the basic tendency for the system to shakedown, if it can. Because of the validity of this shakedown theorem, the problem of designing a system under dynamic loadings and with only a finite amount of allowable plastic work can be transformed into a quasi-static, elastic counterpart. For the case of proportional loadings, two methods for solving the “compounded” shakedown load are proposed. One is called the “Method of Zero Work;” the other, involving a systematic numerical procedure, is called the “Method of Direct Search.” The concept of “optimum preloading” is also introduced.

Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: A finite element, linear programming approach

The matrix description of the mechanical behaviour resting on finite element discretization and piecewise linearization of yield surfaces, is adopted instead of the traditional continuous field description. By the use of linear programming concepts, the essential of a general shakedown theory and the basis of relevant solution procedures are presented in compact form. For systems with associated flow-laws, the second shakedown theorem (Koiter's) is extended in order to allow for variable dislocations (e. g. temperature cycles). For systems with nonassociated flow-laws two theorems are given which supply lower and upper bounds to the safety factor.

Concerning the Further Understanding of the Shakedown Theorems Used for Structural Analysis

Technical Briefs Concerning the Further Understanding of the Shakedown Theorems Used for Structural Analysis Melvin Zaid Melvin Zaid Scientific Research Staff, Republic Aviation Corporation, Farmingdale, N. Y. Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information Melvin Zaid Scientific Research Staff, Republic Aviation Corporation, Farmingdale, N. Y. J. Appl. Mech. Sep 1959, 26(3): 462-463 (2 pages) https://doi.org/10.1115/1.4012070 Published Online: June 23, 2021 Article history Received: January 9, 1959 Published: September 1, 1959 Online: June 23, 2021

Plastic Collapse and Shakedown Theorems for Structures of Strain-Hardening Material

In recent years at tempts have been made to develop methods of design for redundant structures of ductile material which are based upon the calculation of the load a t which a structure actually collapses as a result of excessive plastic deformation. Such methods of plastic or limit design may be used when it is known in advance tha t all the loads on the structure will be applied simultaneously. However, when a structure is subjected to several loads, each of which may vary independently of the other, a different type of failure must be considered. As the loads vary, a cycle of plastic deformations may be established, even though no possible combination of the loads could cause collapse. In this case, failure would occur either (1) by the fracture of some member in which plastic deformations occurred alternately in one sense and then in the other or (2) by the development of excessive plastic deformation as a result of a finite increase in the amount of plastic deformation occurring during each cycle of loading. I t is therefore necessary to ensure that , as the loads vary, the structure attains a state of residual stress such that no further plastic flow can occur, no matter how often and in what order the loads are repeated. When this occurs, the structure is said to have shaken down. Recent theoretical work on the problems of plastic collapse and shakedown has been restricted to the case of trusses and framed structures whose members exhibit the ideal-plastic type of stressstrain relation. In the present paper, some theorems concerning collapse and shakedown are established for trusses and framed structures whose members possess a strain-hardening type of characteristic. The most interesting cases, covered by the theorems, are those in which the strain hardening is limited, so that the stress-strain characteristic eventually becomes horizontal and plastic deformation can continue indefinitely under constant stress. The method adopted for the proof of these theorems is to simulate the strain-hardening characteristic by imagining a large number of bars of ideal-plastic material to be placed in parallel and constrained to have the same elongation. In this way theorems are established which are closely analogous to those tha t apply to structures of ideal-plastic material and which are in a suitable form to enable plastic collapse and shakedown loads to be calculated.