INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2012; 91:896–908 Published online 17 April 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4303 A note on upper bound formulations in limit analysis J. J. Muñoz 1, * ,† , A. Huerta 1 , J. Bonet 2 and J. Peraire 3 1 Dep. Applied Mathematics III, LaCàN, Univ. Poitècnica de Catalunya (UPC), Barcelona, Spain 2 Civil and Computational Engineering Centre, School of Engineering, University of Wales, Swansea, UK 3 Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (MIT), Cambridge, MA, USA SUMMARY In this paper, we study some recent formulations for the computation of upper bounds in limit analysis. We show that a previous formulation presented by the authors does not guarantee the strictness of the upper bound, nor does it provide a velocity field that satisfies the normality rule everywhere. We show that these deficiencies are related to the quadrature employed for the evaluation of the dissipation power. We derive a formulation that furnishes a strict upper bound of the load factor, which in fact coincides with a formulation reported in the literature. From the analysis of these formulations, we propose a post-process, which consists in computing exactly the dissipation power for the optimum upper bound velocity field. This post-process may further reduce the strict upper bound of the load factor in particular situations. Finally, we also deter- mine the quadratures that must be used in the elemental and edge gap contributions, so that they are always positive and their addition equals the global bound gap. Copyright © 2012 John Wiley & Sons, Ltd. Received 5 February 2010; Accepted 19 January 2011 KEY WORDS: limit analysis; bounds; optimisation; numerical integration 1. INTRODUCTION Lower (upper) bounds of the load factor  in limit analysis are obtained by constructing discrete spaces of the velocity and stress fields, v and  , respectively, which are statically (kinematically) admissible. Although there is a common agreement in the choice of statically admissible spaces, several options have been reported when designing kinematically admissible spaces that yield upper bounds of . The objective of this paper is to study the properties of two proposed upper bound formulations, and suggest potential improvements. Kinematically admissible spaces were originally derived by employing piecewise linear and con- tinuous velocity fields. Although some recent (mixed) formulations have been proposed resorting to continuous velocity fields [1, 2], the incompressibility constraints in the velocities for some plasticity criteria does not allow them to furnish tight upper bounds of , unless some specific mesh arrangements are used. To improve the accuracy of these values, velocity discontinuities were originally added in [3] and [4]. These discontinuities were generalised in [5] and [6], and are cur- rently widely exploited [7–12]. Among these references, the formulation in [6, 9] considers some additional velocity variables at the edges, which guarantee the strictness of the upper bound, whereas the more recent article [11] uses quadratic discontinuous velocity fields, which are subjected to some conditions that guarantee its strictness. We also mention the related recent article [13], where the upper bounds of the load factor for multibody structures with frictional contact conditions between them are computed using a mixed linear complementary problem. *Correspondence to: J. J. Muñoz, Dep. Applied Mathematics III, LaCàN, Univ. Poitècnica de Catalunya (UPC), Barcelona, Spain. † E-mail: j.munoz@upc.edu Copyright © 2012 John Wiley & Sons, Ltd.