## 1 Introduction
The stability transformation approach calculated the chaos control factor during the optimization process, thereby enhancing the efficiency of the most probable point search. [PERSON] et al. [15] developed a performance measure approach (PMA) employing the FORM to perform the reliability analysis. In the PMA, the probabilistic constraint was expressed in terms of a performance function, a quantile of the limit-state function. [PERSON] and [PERSON] [16] developed the reliability index approach (RIA), in which the reliability index of the probabilistic constraint was obtained by the FORM. The optimization was performed by updating the design variables to ensure that the obtained reliability index was greater than a predetermined reliability index. [PERSON] et al. [17] proposed an enhanced PMA approach with a hybrid analysis method that adaptively selected the numerical algorithms, average mean value, and conjugate mean value to efficiently obtain the design point that is the most likely failure point. [PERSON] and [PERSON] [18] explored the second-order RIA and PMA formulations with various reliability analysis methods, including the FORM, SORM, mean-value first-order second-moment method, and advanced mean value method. Furthermore, [PERSON] and [PERSON] [19] provided a detailed review of the RIA, [PERSON] et al. [20] conducted a comparative study on the RIA and PMA, and [PERSON] and [PERSON] [21] provided a comprehensive review of the RBDO using the FORM. The sensitivity analysis is an integral part of structural design optimization because the information of the sensitivity analysis indicates the effects of the changes in the design or modeling parameters on the structure. The key variables can be identified through the sensitivity analysis to enable the effective design of different types of structures. Furthermore, the sensitivity information can be used in gradient-based optimization algorithms to search for directions to update the design variables at each iteration. In the context of the RBDO, diverse methods of sensitivity analysis have been developed. [PERSON] et al. [22] proposed a method to compute the sensitivity of the parallel, series, and general system using the sequential compounding method [23], where component events coupled by a union or intersection operation were sequentially compounded. The sensitivity analysis was performed with a single compound representing the system failure to improve the computational efficiency. [PERSON] [24] proposed a simulation approach to perform reliability-based design sensitivity analysis. Specifically, an augmented reliability problem was formulatedusing design variables that were artificially considered as uncertain with a specified probability density function. The reliability sensitivity was obtained by performing the failure analysis of the augmented problem. [PERSON] and [PERSON] [25] utilized a finite difference method for sensitivity estimation within the FORM and SORM. [PERSON] [26] proposed a feasible direction concept to obtain the sensitivity. By changing a selected parameter while keeping other variables fixed, the rate of change in the optimum design was measured. The variation in the optimum solutions was quantified in the sensitivity analysis. Generally, finite difference approaches or simulation-based methods such as Monte Carlo simulations (MCS) may cause the instability of the numerical implementation and require high computational power, and these aspects hinder the application of these techniques. Moreover, the difficulty of sensitivity analysis may increase when the underlying functions and the governing equations of the system are complex, or the analytical derivation of gradients is not feasible. Therefore, to identify the optimal structure that satisfies a certain level of reliability, it is crucial to perform the sensitivity analysis of the failure probability estimated using the SORM, which is generally more accurate than that obtained using the FORM.