## 3 Optimal Design Method Using LM Index
### Formulation of Optimization Problem Using LM Index  
In the optimization process of the material quantity, the cross-sectional areas of the steel beam are obtained through iterative analysis to minimize the objective function under the safety-related constraint conditions. In the GBG system, since the connections between the main beam and girder are modeled as an ideal pinned or rigid connection, the changes in the steel members do not affect the connection modeling. In the DBO system, however, the changes in the sectional properties of the steel double-beam cause the modification in the thickness of the concrete panel. In this study, by modifying the objective function shown in Equation (7), an optimal design method for the DBO system was developed without iterative analysis. By multiplying the unbraced length of each member (\(L^{i}_{b}\)) to the denominator and numerator on the left side of Equation (9), Equation (10) is yielded. Herein, \(L_{b}M_{n}\) is named as the LM index. Figure 6 shows the LM index for hot-rolled H-beam according to the unbraced length. Since the LM index has a unique characteristic of having a maximum value in \(L_{r}\), it can be defined as an index representing the properties of the H-beam. Using this unique characteristic, the LM index in terms of capacity (\(LM_{C}\)) is defined as Equation (11). The LM index in terms of demand (\(LM_{D}\)) is defined as Equation (12). \[\frac{L^{i}_{b}M^{i}_{u}}{\phi_{b}L^{i}_{b}M^{i}_{n}}\leq 1.0 \tag{10}\]  
\[L^{i}_{b}M^{i}_{n}\leq L^{i}_{r}M^{i}_{r}=LM^{i}_{C} \tag{11}\]  
\[L^{i}_{b}M^{i}_{n}=LM^{i}_{D} \tag{12}\]  
where \(L^{i}_{b}\) is the unbraced length of the steel beams (m), \(L^{i}_{r}\) is the limiting laterally unbraced length for the limit state of inelastic lateral-torsional buckling (m), \(M^{i}_{r}\) is the inelastic bending moment for lateral-torsional buckling (kNm), \(LM^{i}_{C}\) is the LM index in terms of capacity (kNm\({}^{2}\)), and \(LM^{i}_{D}\) is the LM index in terms of demand (kNm\({}^{2}\)). The superscript \(i\) denotes \(i\)-th steel member. In KS D 3502 (Korean Industrial Standards) [31], the sectional information, such as unit weight, cross-sectional area, and moment of inertia so on, is presented for 95 H-beams. Figure 7 shows the unit weight for 95 H-beams suggested in KS D 3502 in accordance with the cross-sectional area and the LM index in terms of capacity (\(LM_{C}\)). Since \(LM_{C}\) is defined as the product of \(L_{r}\) and \(M_{r}\) as shown in Equation (11), \(L_{r}\) and \(M_{r}\) are calculated from the  
Figure 6: Relationship between unbraced length (\(L_{b}\)) and LM index (\(L_{b}M_{n}\)) in hot-rolled H-beam. cross-sectional information of KS D 3502 to derive \(LM_{\text{C}}\) for 95 H-beams. As shown in Figure 7a, the cross-sectional area of the H-beam shows a linear proportional relationship to the weight, so it is suitable as an objective function for optimizing the weight. Since the relationship between the \(LM_{\text{C}}\) and the weight presents a proportional relation as shown in Figure 7b, the values of \(LM_{\text{C}}\) can be considered as a weight. If the difference between \(LM_{\text{C}}\) and \(LM_{D}\) is minimized, it is possible to minimize weight as a result. Therefore, the objective function for minimizing material quantity using the LM index as a design variable can be defined as Equation (13). \[Minimize\ W_{LM}=\sum_{i=1}^{M}\Bigl{\{}\phi_{b}LM_{\text{C}}^{i}-LM_{D}^{i} \Bigr{\}} \tag{13}\]  
where \(W_{LM}\) is the objective function of the material quantity expressed in the LM index (kNm2). The constraint conditions used in the proposed optimal design method using the LM index are expressed as Equations (14) and (15). In the case of optimization using only the LM index, an increment in the unbraced length leads to a decrement in the nominal flexural strength, so the safety-related design condition may not be satisfied. Equation (14) is employed as a safety-related constraint condition in the proposed optimal design method to prevent violation of the design condition.