popular, but it is computationally expensive and sometimes troubled by convergence

problems, etc. (Tu et. al. 2000). The computational cost of RBDO with nested MPP may

be reduced by sensitivity analysis. The sensitivity of the safety index to design variables

can be obtained with little extra computation as by-products of reliability analysis (Kwak

and Lee 1987). A simplified formula that ignores the higher order terms in the estimation

equation was proposed by Sorensen (1987). Yang and Nikolaidis (1991) used this

sensitivity analysis and optimized an aircraft wing with FORM subjected to system

reliability constraint.

 Figure 2-1 shows the typical procedure for the double loop approach. With this

approach, the reliability constraints are approximated at the current design point (DP) dk.

 For problems requiring expensive finite element analysis, this approach may still be

computationally prohibitive; and FORM (e.g., classical FORM such as Hasofer-Lind

method) may converge very slowly (Rackwitz 2000). Wang and Grandhi (1994)

developed an efficient safety index calculation procedure for RBDO that expands limit

state function in terms of intermediate design variables to obtain more accurate

approximation. Reliability constraints can also be approximated to reduce the

computational cost of RBDO. Wang and Grandhi (1994) approximate reliability

constraints with multi-point splines within a double loop RBDO. Another way of

improving the efficiency of multi-level optimization is to integrate the iterative

procedures of reliability analysis and design optimization into one where the iterative

reliability analysis stops before full convergence at each step of the optimization, as

suggested by Haftka (1989). Maglaras and Nikolaidis (1990) proposed an integrated

analysis and design approach for stochastic optimization, where reliability constraints are