## 3 Materials and Methods
### AI-Based Optimization
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Dynamo Nodes** & **RSA API** & **Python Library** \\  & [8] & [34] & **This Study** \\ \hline Required Time (Min. Approximately) & 240 & 30 & 0.4 \\ Number of FEM Evaluation & 200 & 200 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time required for FEM analyses. are word steel sections for the frame's columns and beams, making the example a discrete RBDO problem. A list containing 36 different W14 column profiles and 16 different W21 beam profiles for each design variable is sourced from the AISC LRFD Manual [73] for optimization. The objective function of this optimization is to minimize the structural volume while ensuring that the inter-story drift does not exceed 0.01 m, which is in line with three defined reliability constraints for all stories. In the final example, a 3D structure is sourced from referenced article [75] which proposed a new standard benchmarking suite called \"ISCSO 2016-2022\" for evaluating structural optimization algorithms using problems from the International Student Competition in Structural Optimization (ISCSO) [76]. The authors aim to address the lack of challenging modern benchmarks in the field. Each benchmark problem is formulated to minimize the mass of truss structures while adhering to constraints on strength and displacement based on AISC-LRFD standards [77]. The paper and website provide detailed formulations for different test cases, ensuring consistency and accuracy in comparison. The ISCSO 2019 problem, which has 260 truss members and 270 design variables, presented in Figure 7c, has been adopted for this study. However, to optimize it within a reasonable computational time, the problem has been scaled by using three nodes in the x direction instead of 19. This new 36-member space truss structure, shown in Figure 7d, represents a complex three-dimensional structural system commonly used in industrial applications. The cross-sectional areas of the members are selected from a discrete set of 37 standard pipe sections sourced from [77], which could also be found in [76]. The loading conditions include three independent forces: P\({}_{1}\) and P

Figure 7: Structures chosen for the results section are as follows: (**a**) 10-bar truss structure; (**b**) 2D frame structure; (**c**) ISCSO 2019 problem; and (**d**) 3D-36 bar truss structure. with mean values of 50 kN acting in the x and y directions, respectively, and P\({}_{3}\) with a mean value of 10 kN acting in the negative z direction. The material properties are characterized by a modulus of elasticity (E) with a mean value of 200,000,000 kN/m\({}^{2}\). The optimization problem considers both sizing variables (cross-sections for all members) and shape variables (z-coordinates of bottom nodes) with bounds of \(-\)2500 mm to 3500 mm, making it a mixed discrete-continuous RBDD problem with 38 random variables. For reliability analysis, all random variables (P\({}_{1}\), P\({}_{2}\), P\({}_{3}\), and E) follow lognormal distributions, while the y coordinates follow a normal distribution with a standard deviation of 0.05. The objective function is the total mass of the structure. The LSF is formulated based on the maximum nodal displacement, with a constraint of 2.5 mm in any direction. This ensures the structure maintains its serviceability under the uncertain loading and material conditions. The reliability requirement is specified with a target reliability index (\(\beta\)) of 3.0, corresponding to a probability of failure of approximately 0.00135. ### Results and Analysis

For the results section, the structures are optimized with the constraints and configurations explained in the previous section. Python scripts have been prepared to compute the MCS to provide and verify the reliability of optimized structures. Additionally, results from structural optimizations found in the literature have been used to compare the reliability of these structures with those optimized using RBDO in this study.