Coupled Systems Mechanics, Vol. 2, No. 4 (2013) 349-374 DOI: http://dx.doi.org/10.12989/csm.2013.2.4.349 349 Copyright © 2013 Techno-Press, Ltd. http://www.techno-press.org/?journal=csm&subpage=7 ISSN: 2234-2184 (Print), 2234-2192 (Online) Linear instability or buckling problems for mechanical and coupled thermomechanical extreme conditions Adnan Ibrahimbegovic 1 , Emina Hajdo 1,2 and Samir Dolarevic 2 1 Laboratoire de mécanique et technologie, École Normale Supérieure, 61 Avenue du Président Wilson, 94230 Cachan, France 2 Faculty of Civil Engineering, University of Sarajevo, Patriotske lige 30, 71000 Sarajevo, Bosnia and Herzegovina (Received January 30, 2014, Revised March 11, 2014, Accepted March 12, 2014) Abstract. In this work we propose a novel procedure for direct computation of buckling loads for extreme mechanical or thermomechanical conditions. The procedure efficiency is built upon the von Karmann strain measure providing the special format of the tangent stiffness matrix, leading to a general linear eigenvalue problem for critical load multiplier estimates. The proposal is illustrated on a number of validation examples, along with more complex examples of interest for practical applications. The comparison is also made against a more complex computational procedure based upon the finite strain elasticity, as well as against a more refined model using the frame elements. All these results confirm a very satisfying performance of the proposed methodology. Keywords: buckling; thermomechanical coupling; geometric instability, critical load, eigenvalue problem 1. Introduction One of the most frequent cases of extreme loading conditions that engineering structures can be exposed to, pertains to fire. The most sadly famous example, the World Trade Center collapse in September 2001, is certainly not the only one. One can cite a number of catastrophic failures under combined action of mechanical and thermal loads, such as MGM Grand Hotel in 1980 in Nevada, First Interstate Bank in Los Angeles in 1988 or Windsor Tower in Madrid in 2005. Each of these failures has been caused by the same mechanism of combined action of mechanical loading and high temperature. Hence, for the predictive analysis of the phenomena of this kind we ought to provide the correct representations of two different failure mechanisms. The first one pertains to softening material failure under temperature increase (e.g., Van Ngo et al. (2013)), and the second one pertains to geometric instability phenomena brought about the critical values of mechanical and thermal load. The latter failure mechanism pertaining to buckling and its correct representation are studied in detail in this work. In particular, we seek to provide as simple as possible and yet sufficiently predictive model for Corresponding author, Professor, E-mail: adnan.ibrahimbegovic@ens-cachan.fr