## 5 Numerical Examples
### The 18-Bar Planar Truss  
The second benchmark example is an 18-bar planar truss structure involving continuous variables (Figure 8). The modulus of elasticity is \(10^{4}\) ksi and the material density is 0.1 lb/in\({}^{3}\). The structure is subjected to five concentrated loads (P), each applying a downward force of 20 kips to nodes 1, 2, 4, 6, and 8. Figure 6: The optimized shapes of the 15-bar truss with a color map of stresses in members and a detailed view of nodes 4 and 8 by (**a**) CSBO and (**b**) NCSBO. Figure 7: Comparison of convergence curves for the 15-bar truss. This truss example involves a total of twelve design variables, with four for sizing optimization and eight for shape optimization, as outlined below:  
\[ \begin{array}{c}\textit{Sizing variables}\\ A=\begin{bmatrix}A_{1}=A_{4}&=A_{8}&=A_{12}&=A_{16}\\ A_{5}=A_{9}&=A_{13}&=A_{17}\\ A_{2}&=A_{6}&=A_{10}&=A_{14}&=A_{18}\\ A_{3}&=A_{7}&=A_{11}&=A_{15}\end{bmatrix}_{nsize\times 1}\\ \textit{Layout variables}\\ X=[x_{3},x_{5},\ x_{7},x_{9},\ y_{3},y_{5},y_{7},y_{9}]\end{array} \]  
The boundary conditions of layout variables are considered as given below:  
\[ \begin{array}{c}775\leq x_{3}\leq 1225(in);\ 525\leq x_{5}\leq 975(in);275\leq x _{7}\leq 725(in);\\ 25\leq x_{9}\leq 475(in);-225\leq y_{3},y_{5},y_{7},y_{9}\leq 245(in);\end{array} \]  
The allowable cross-sections for sizing variables are confined to the discrete profile set outlined as =[2, 2.25, 2.5, \(\ldots\) 21.25, 21.5, 21.75] (in2). The maximum stress for all members must not exceed \(\pm 20\) ksi. The allowable buckling stress is considered a design constraint and calculated as \(4 EA/L^{2}\) for each member subjected to compression, in which \(E\) is the modulus of elasticity, \(A\) denotes the cross-sectional area, and \(L\) is the member length. Table 6 summarizes the optimization results obtained by CSBO, NCSBO, and the literature. NCSBO shows the best performance, achieving the lightest design at 4512.1244 lb after 5900 structural analyses. The optimal design found by CSBO, weighing 4530.13 lb after 7585 analyses, also shows strong performance relative to other methods. Furthermore, all solutions were carefully checked for constraint violations, and it was found that the allowable stress value for tension members in the MLA was 25 ksi instead of 20 ksi. As a result, a direct comparison between the optimized design obtained by the MLA and other methods found in the literature is not possible. Upon re-evaluating the design produced by the MLA against the 20 ksi stress limit, it was determined that it exceeded the design constraints by 24.87%. NCSBO shows the best convergence capability after ABC; however, NCSBO yielded the optimized results of ABC after 5150 vs. 2700 structural analyses. The STD for ten independent values is calculated at 5.39 lb, which is about 0.12% of the mean weight, as reported in Table 6. This result verifies the consistency of designs obtained by independent runs found by NCSBO. Figure 9 illustrates the distribution of stresses in the members according to maximum, minimum, and buckling limit stresses. In the optimal design obtained with NCSBO, tensile stresses in members 16 and 17 are recorded as 19.999 ksi and 19.9113 ksi, respectively, approaching the maximum allowable tensile stress of 20 ksi. Additionally, members 2, 6, 7, 10, 11, and 14 are subjected to compressive forces closely approaching their buckling  
Figure 8: The 18-bar planar truss. limit values. Similarly, in the CSBO design, members 16 and 17 exhibit tensile stresses of 19.997 ksi and 19.932 ksi, respectively, nearing the tensile stress limit.