Anti-optimisation of uncertain structures using interval analysis Stewart McWilliam * School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, University Park, Nottingham NG7 2RD, UK Received 1 November 1999; accepted 1 May 2000 Abstract This paper considers the static displacement bounds of structures modelled using uncertain (but non-random) para- meters. In the analysis, the uncertain parameters are assumed to take deterministic values within a speci®ed interval, and the bounds of the displacement are obtained by solving interval linear equations. Two new methods for solving these equations are proposed. The ®rst is a modi®ed version of interval perturbation analysis, while the second is based on the assumption that the displacement surface is monotonic. Numerical results indicate that both methods are more accurate than standard interval perturbation analysis and more ecient than the combinatorial approach. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Anti-optimisation; Uncertain-but-non-random; Static; Interval perturbation analysis; Combinatorial approach 1. Introduction In the design of engineering structures, it is usual to assess their behaviour using speci®c values for structural parameters and applied loads. In practice, however, there is always some degree of uncertainty associated with the actual values for structural parameters and loads. A consequence of this is that the response of the structure will always exhibit some degree of uncertainty. Until recently, probabilistic methods have formed one of the only ways of taking account of uncertainties. How- ever, given that only a small amount of statistical in- formation about structural parameters and loads is known in a few specialised cases, there has been in- creased interest in the application of models of uncer- tainty which do not depend upon such detailed knowledge. Ben-Haim and Elishako [1] and Ben-Haim [2] have proposed a bounded uncertainty approach for this purpose which relies upon knowledge of the bounds of uncertainty. In this approach, the strategy adopted is that of Ôanti-optimisationÕ in which the least favourable response under the imposed constraints is determined. In practical applications of this approach, it is usual to use (elliptical) convex sets to model the uncertain phe- nomena, while more recent work has used interval sets [3±11]. The present work considers the latter. Qiu et al. [3±5] and Rao [6±8] have considered the application of interval methods in the study of the static displacement and eigenvalues of structures with uncer- tain parameters. Under the assumption of small varia- tions about the nominal parameter value, Qiu used a ®rst-order interval perturbation approach to determine the in¯uence of interval parameters on eigenvalues [3] and static displacement [4,5] of structures. Whilst indi- cative of the magnitude of the displacement bound, this method neglects certain interactions that exist between parameters within the associated stiness matrix and force vector. As a result, a degree of uncertainty remains about the absolute accuracy of the calculation. Rao et al. [6±8] has investigated a combinatorial approach for determining the least favourable response. In this Computers and Structures 79 (2001) 421±430 www.elsevier.com/locate/compstruc * Tel.: +44-115-9513772; fax: +44-115-9513800. E-mail address: stewart.mcwilliam@nottingham.ac.uk (S. McWilliam). 0045-7949/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII:S0045-7949(00)00143-7