## 1 Introduction  
Structural design is primarily aimed at maximizing the strength, serviceability, and safety of a structure considering the load effects. The dimensions, compositions, and orientations of the structural members are carefully determined to ensure a sufficient safety margin between the structural demand and its capacity. However, uncertainties inevitably arise in the planning and execution of processes in structural analyses, modeling, design parameters, and strength evaluations. In particular, the environmental conditions, geometric characteristics, and nature of material compositions, along with discrepancies between the predicted and actual loads, responses, and strength, add many layers of complexity and uncertainty. Consequently, in structural design and optimization, the uncertainties in predicting the load effects and structural modeling must be considered to ensure a sufficient level of structural reliability. Nevertheless, this consideration is challenging because in addition to the increasing number of natural and artificial uncertainties, the nature of their interaction also becomes unpredictable and is difficult to anticipate. To address the ongoing challenges associated with increasing uncertainties, theoretical and numerical probabilistic analysis and design methods are being developed in academia and industry. The reliability-based design optimization (RBDO) strategy incorporates the reliability analysis to account for uncertainties in structural design optimization. The RBDO framework includes the evaluation of the probabilistic constraints, implemented through the reliability analysis during optimization. The reliability analysis is aimed at determining the probability that a component or a system will fail in operation, considering the impacts of random inputs. However, the estimation of the failure probability of a structure is generally not straightforward and is therefore computationally expensive. Many algorithms and methods have been proposed to address these challenges and increase the efficacy of the reliability analysis, including simulation-based [1; 2], surrogate model [3; 4], matrix-based [5], first-order linearization [6; 7], and quadratic approximation approaches [8; 9; 10]. Among these techniques, the first-order reliability method (FORM) linearizes the limit-state equations to approximate the probability, whereas the second-order reliability method (SORM) uses a quadratic approximation to perform the reliability analysis [11; 12]. The estimation accuracy depends on the characteristics of the limit-state function. When the limit-state function has a high nonlinearity, the SORM achieves more accurate evaluations compared to the FORM that may not identify accurate failure points or encounter a convergence issue while searching for them. Various factors, such as the nonlinear relationship between random variables and the transformation of random variables from the original space to standard normal space, may cause the nonlinearity of the limit-state function. In contrast, when the limit-state function is linear (or close to linear) in the standard normal space, the FORM and SORM may generate equivalent calculation results for the probability. The FORM and SORM are being increasingly used owing to their high computational efficiency, accuracy, and wide applicability. [PERSON] and [PERSON] [13] proposed an RBDO technique to minimize the radius of roadway horizontal curves by performing the first-order reliability analysis. Analytical models integrated with the vehicle behaviors and characteristics were used to establish the probabilistic constraints, and novel recommendations of the minimum radii were provided to satisfy the target reliability levels. [PERSON] et al. [14] proposed an RBDO technique using the SORM with an improved stability transformation approach to increase the accuracy and efficiency of the reliability analysis.