Nuclear Engineering and Design 237 (2007) 1231–1240 Winner of Jaeger Prize SMiRT – 18 A simplified technique for shakedown limit load determination Hany F. Abdalla a,∗ , Mohammad M. Megahed a,1 , Maher Y.A. Younan b,2 a Department of Mechanical Design and Production, Faculty of Engineering, Cairo University, Egypt b Mechanical Engineering Department, The American University in Cairo, Egypt Received 25 February 2006; received in revised form 21 September 2006; accepted 25 September 2006 Abstract In this paper, a simplified technique is presented to determine the shakedown limit load of a structure using the finite element method. The simplified technique determines the shakedown limit load without performing lengthy time consuming full elastic-plastic cyclic loading simulations or conventional iterative elastic techniques. Instead, the shakedown limit load is determined by performing two analyses namely: an elastic analysis and an elastic-plastic analysis. By extracting the results of the two analyses, the shakedown limit load is determined through the calculation of the residual stresses developed within the structure. The simplified technique is applied and verified using two bench mark shakedown problems namely: the two-bar structure subjected to constant axial force and cyclic thermal loading, and the Bree cylinder subjected to constant internal pressure and cyclic high temperature variation across its wall. The results of the simplified technique showed very good correlation with the, analytically determined, Bree diagrams of both structures. In order to gain confidence in the simplified technique, the shakedown limit loads output by the simplified technique are used to perform full elastic-plastic cyclic loading simulations to check for shakedown behavior of both structures. © 2006 Elsevier B.V. All rights reserved. 1. Introduction The term shakedown was initially introduced into the context of solid mechanics by Melan (1938a,b) through the shakedown theorem stated as follows: “For a given load set P, if any dis- tribution of self-equilibrating residual stresses can be found (assuming perfect plasticity) which, when taken together with elastically calculated stresses, constitute a system of stresses within the yield limit, then P is a lower bound shakedown load set and the structure will shakedown”. During the last three to four decades, research efforts have been focused on limit load analyses in order to determine the load carrying capac- ity a structure or a component can withstand prior to collapse. Complementary to determining limit load, the safe operating region of a structure subjected to cyclic loading within the elastic-plastic domain, the shakedown domain, should be deter- mined as well to avoid early failure due to reversed plasticity ∗ Corresponding author. Fax: +202 795 7565. E-mail addresses: hany f@aucegypt.edu (H.F. Abdalla), mmegahed47@yahoo.com (M.M. Megahed), myounan@aucegypt.edu (M.Y.A. Younan). 1 Fax: +202 570 3620. 2 Fax: +202 795 7565. or ratchetting. Operation under reversed plasticity conditions causes early failure due to low cycle fatigue while operation under ratchetting conditions causes failure due to incremen- tal accumulation of plastic strains which may exhaust material ductility. Due to the high expenses of experimental setups and the time consuming cyclic elastic-plastic finite element analyses, the determination of structural responses under cyclic load- ing conditions are delayed compared with the achievements in limit load analyses. However, with the growing computing powers currently witnessed the elastic-plastic analyses become more reliable, economical, and yield more precise solutions than before. Hence, research efforts are focused on developing simplified numerical techniques capable of economically deter- mining suitable bounding solutions to a variety of shakedown problems. 2. Literature review Despite the introduction of the shakedown theorem by Melan (1938a,b) in the late 1930s, active research in this area began in the mid-1960s. Most of the work accomplished aimed at determining the shakedown loads and working domains for cyclically loaded structures focusing on pressure vessels (Leckie 0029-5493/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2006.09.033