1 INTRODUCTION The main objective of design optimization is to ob- tain a set of design variables that minim- ize/maximize objective function(s) of interest while satisfying given constraints. If design optimization is performed in a deterministic manner, i.e. uncertain- ties are not taken into account during the optimiza- tion, the resultant optimal design may have unquan- tified risk of violating the given constraints. Recently, various reliability based design optimiza- tion (RBDO) methods have been developed to achieve optimal designs with acceptable failure probabilities (see Frangopol & Maute 2004 for a state-of-the-art review of RBDO and recent applica- tions to civil and aerospace structural systems). Dur- ing RBDO, the probability of violating given con- straint(s), i.e. the failure probability, is often computed by reliability analysis employing methods such as first order reliability method (FORM), second order reliability method (SORM) or response surface method. Traditionally, RBDO has been performed by use of a nested or “double loop” approach, that is, each step of the iteration for design optimization involves another loop of iteration for reliability analysis. For example, reliability index approach (RIA; Enevold- sen & Sorensen 1994) and performance measure ap- proach (PMA; Tu et al. 1999) employ FORM as the inner loop to perform the reliability analysis effi- ciently. If the constraints are active, the two ap- proaches yield the same results. However, it is known that PMA is generally more efficient and sta- ble than RIA (Tu et al. 1999). In general, the double loop approach is computa- tionally expensive. Recently, a single-loop approach (Liang et al. 2004) was proposed to improve effi- ciency of RBDO. The Karush-Kuhn-Tucker (KKT) optimality condition is used to approximate the de- sign point (or most probable point, MPP) in the in- ner loop for each constraint. As a result, the inner loop is replaced by a deterministic constraint, which transforms the double loop RBDO problem into an equivalent deterministic optimization problem. When multiple failure modes need to be consi- dered as the constraints of a design optimization, RBDO is often formulated such that the optimal structure satisfies each failure mode with pre- determined probabilities. This approach is termed as component reliability-based design optimization (CRBDO) in this paper. In some cases, however, the failure event is better described by a system event, i.e. a logical (or Boolean) function of multiple fail- ure modes. In this case, the probabilistic constraint should be given for the system event. This approach is called system reliability-based design optimiza- tion (SRBDO). The SRBDO requires system relia- bility analysis, which is not trivial especially for sta- tistically dependent component events, or for a system event that is not series or parallel system. Single-Loop System Reliability Based Design Optimization (SRBDO) Using Matrix-based System Reliability (MSR) Method T.H. Nguyen, J. Song, and G.H. Paulino University of Illinois, Urbana, Illinois, USA ABSTRACT: This paper proposes a single-loop system reliability based design optimization (SRBDO) ap- proach using the recently developed matrix-based system reliability (MSR) method. A single-loop method was employed to eliminate the inner loop of SRBDO that evaluates probabilistic constraints. The MSR me- thod computes the system failure probability and its parameter sensitivities efficiently and accurately through efficient matrix calculations. The SRBDO/MSR approach proposed in this paper is uniformly applicable to general systems including series, parallel, cut-set and link-set system events. Two numerical examples dem- onstrate the proposed approach. In the first example, the cross-sectional areas of the members of a statistically indeterminate truss structure are determined for minimum total weight with a constraint on the system failure probability satisfied. The second example demonstrates the application of the proposed approach to topology optimization. The influences of the statistical correlation and the types of constraints, i.e. deterministic, prob- abilistic (component) and probabilistic (system) on the optimal topology are investigated. Safety, Reliability and Risk of Structures, Infrastructures and Engineering Systems – Furuta, Frangopol & Shinozuka (eds) © 2010Taylor & Francis Group, London, ISBN 978-0-415-47557-0 1534