Limit Analysis of Structures at Thermal Cycling

Abstract

Reviewed by Dr. John ChristodoulidesREVIEW -The methods of elastic-plastic analysis of continua were developed in response to increased efficiency and reliability requirements on modern structural components. The fundamentals of the theory of plasticity have already been implemented in a number of available computer codes. Among the methods of plastic analysis of structures, the theory of limit analysis has already become an engineering tool. The method employs the rigid-plastic model of material behavior and it is capable fo evaluating upper and lower bounds to the ultimate load intensity that can be sustained by a structure. If variable repeated loading is present, a suitable generalization of the limit analysis is introduced, known as the shakedown theory. Initial development of the shakedown theory dates back to the thirties and the shakedown analysis of structures under variable temperature fields were initiated by Prager in the fifties. The present text is among the rare comprehensive treatments of elastic-plastic structures under cyclic mechnaical and thermal loading.The concept of elastic-plastic response under variable repeated loading and temperature is first introduced herein for a simple bar system. The possible ensuing regimes of a stabilized response or an incremental collapse process are depicted in order to enhance reader understanding of the subject matter.All the concepts of applied mechanics pertinent to the contents are illustrated before the theorems of shakedown are stated and proved. Analogies existing between shakedown analysis and limit analysis are presented. The generalized stresses and strains used traditionally in plate and shell analysis are used and the deviation of limiting interaction surfaces for variable cyclic stresses is illustrated.The solution of shakedown problems can be obtained with the aid of mathematical programming techniques. The mathematical background in the involved process is outlined and detailed examples are given for simple structural elements under cyclic mechanical and thermal loading.The book devotes considerable attention to the analysis of rotating disks at cyclic variations of angular velocity and temperature. Shakedown diagrams are derived for a number of plate and shell problems, and thus their capacity under cyclic pressure and temperature profiles is identified. Progressive thermal cycling deformation (ratchetting) and contact problems are included. Finally theoretical proof of the existence and uniqueness of the elastic stabilization (shakedown) state is given.A wealth of information concerning elastic-plastic analysis of structural components under thermal and mechanical cycling is given in this monograph. It is concise, complete and it is considered extremely valuable in its field for both research and design.

Kinematical shakedown analysis with temperature-dependent yield stress

“…It is recognized that the plastic fatigue limit may be determined by the condition that anywhere in structures, the maximum varying magnitude of ÿctitious equivalent elastic stress cannot exceed two times of the yield limit of material [9]. So the alternating plasticity limit may be found based only on an elastic computation.…”

“…The load domain may be described by T ⊂ (0; 1)T and p ⊂ (0; 1)p or p ⊂ (1; 1)p. Owing to the central symmetry, only one-fourth of the sphere is modelled by axisymmetric quadratic ÿnite elements. The analytic shakedown limit is presented as Equations (43) and (44) [9] for the comparison:…”

“…Large amounts of studies have been carried out since the investigations initiated by Prager in 1956 [8]. An extensive treatise on this subject was published by Gokhfeld and Cherniavsky [9]. Numerical studies of particular problems were presented by Goodman [10], Leckie and Penny [11], Sagar and Payne [12], Ponter [13], Kleiber and K onig [14], Weichert and GrossWeege [15], Maier et al [2] and other authors.…”

FEM-based shakedown analysis of hardening structures

“…The unbounded kinematic hardening model cannot estimate the plastic collapse and also incremental plasticity but only low-cycle fatigue, while low-cycle fatigue limit with the kinematical hardening model seems not to be essentially different from the perfectly plastic model, cf. Gokhfeld and Cherniavsky [5] and Stein and Huang [6].…”

Shakedown working limits for circular shafts and helical springs subjected to fluctuating dynamic loads

“…Under such fluctuating dynamic loads, a structure would not collapse instantaneously according to the classical plastic limit theory, thanks to the inertia effect, but would fail incrementally (ratcheting mode) or by alternating plasticity (fatigue mode). The problem can be solved in the framework of shakedown theory [Koiter 1963;Gokhfeld and Cherniavski 1980;König 1987;Bree 1989;Pham 1992;2005;2007;2008;Pham and Stumpf 1994;Pham and Weichert 2001;Weichert and Maier 2002].…”