## 1 Introduction
The reliability index approach (RIA) [9] and its inverse, the performance measure approach (PMA) [10; 12], utilize the first-order reliability method (FORM) [13] for reliability analysis. RBDO is performed using a nested approach, such that each iteration in the design optimization process involves a sub-loop of iterations for reliability assessment at the current stage of design. This is often called a \"double-loop\" approach because each step of the optimization process involves another iteration for the reliability analysis [13]. As a result of the repeated iteration process, the computational cost associated with the utilization of the double-loop approach can prohibitively be expensive. To improve the computational efficiency of reliability analysis and optimization procedure, many RBDO approaches, such as decoupling [14; 15] and single-loop (SL) methods [16; 17; 18; 19; 20], have been proposed. The decoupling method decouples the RBDO procedure by constructing a sequence of cycles with deterministic optimization followed by the reliability analysis of the converged solution in the feasible domain. The SL approach is based on the Karush-Kuhn-Tucker (KKT) optimality conditions to approximate the solution of the sub-loop optimization. As a result, the sub-loop for reliability analysis is replaced by a deterministic constraint, which transforms a double-loop RBDO problem into an equivalent single-loop optimization problem. RBDO methods have been applied to many fields, including structural engineering [17; 21], mechanical engineering [22; 23], aerospace engineering [24], and material science [25]. A comprehensive review on RBDO can be found in the work of [PERSON] et al. [26], [PERSON] and [PERSON] [27], and [PERSON] et al. [28]. In DDO, RDO, and RBDO, sensitivity analysis of probabilistic constraints is crucial to the utilization of efficient optimization algorithms. The dependence on reliability with respect to the design inputs or values can be evaluated using sensitivity analysis. There have been multiple approaches in sensitivity analysis, such as Monte Carlo Simulation-based approaches [29; 30], and simulation-based methods [31; 32]. However, these approaches generally require high computational power that might restrict their applicability, especially when sensitivity analysis needs to be performed with respect to many parameters. The difficulty of sensitivity analysis increases when the underlying functions and models are complex, or the analytical derivation of gradients is not feasible due to the implicit system. In this paper, the sensitivity analysis of failure probability, a probabilistic constraint of RBDO, is proposed using complex-step (CS) approximation integrated with the FORM. [PERSON] [33] showed sensitivity analysis in the FORM-based RBDO using the CS. This paper provides details on the general integration of CS in the FORM and discusses uniform applicability to structural RBDO. The sensitivity analysis is performed using a very small complex value, such that the outcome is not significantly affected by a selection of steps, unlike the finite difference method. In addition, the proposed method can be easily implemented in RBDO and is free from round-off and subtractive cancellation errors when compared with the finite difference method. Furthermore, sensitivity analysis and assessment of the failure probability can be simultaneously performed using the CS approximation simply by taking real and imaginary parts of numerical results so that computational efficiency in analysis and optimization increases. This paper presents the RBDO method utilizing the FORM and presents a sensitivity analysis of probabilistic constraints with respect to design and random variable parameters to use a gradient-based optimization algorithm.