# Evaluation of Link Overstrength Factor for the Seismic Design of Eccentrically Braced Frames
## 3 Design of Example Buildings
### Generation of Design Alternatives Using Multi-Objective Optimization  
In general, structural members are selected to ensure safety and economic efficiency. Therefore, the optimal solution may not be unique. Furthermore, even under the same design conditions, the selection of structural members may vary depending on the designer's preferences. Multi-objective optimization techniques, which generate multiple optimal solutions, can be effective numerical procedures for accounting for this diversity in the design process. In this study, a multi-objective metaheuristic algorithm based on NSGA-II (nondominated sorting genetic algorithm) was employed to generate multiple and varied design examples. The two objective functions used in the optimization are expressed in Equation (8a,b). The first objective function minimizes the sum of structural members. The second objective function maximizes the average value of the maximum demand-to-capacity ratios (DCRs) in member groups. The relevant constraints are appended, as shown in Equation (8c), which includes the condition that the demand strength cannot exceed the capacity of individual members, the width-to-thickness ratio requirement of the section, the link plastic rotation limitations, and the inter-story drift. Additionally, the conventional practice, in which the sections of columns on the lower story cannot be smaller than those on the upper story, was incorporated. Minimize \[f_{1}=\sum\limits_{i=1}^{N_{slt}}wA_{i}L_{i}+\sum\limits_{j=1}^{N_{slt}}wV_{ stf,j}\] (8a)Maximize \[f_{2}=\left[\frac{1}{N_{grp}}\sum\limits_{i=1}^{N_{grp}}\left( \max\limits_{j}\frac{R_{u,j}}{\phi R_{n,j}}\right)_{i}\right]\] (8b)Subject to \[ \begin{array}{c}R_{u}\leq\phi R_{n}\\ b/t\leq(b/t)_{limit}\\ \theta_{p}\leq\theta_{pa}\\ \Delta_{d}/h\leq\Delta_{a}/h\\ d_{c,i}\leq d_{c,i-1}\end{array} \hskip 56.905512 pt \begin{array}{c} \text{herein, }\Delta_{d}=\Delta_{e}C_{d}/I_{E}\\ i=2,\ldots,N_{story}\end{array} \] (8c)  
where \(N_{ele},N_{slt}\), \(N_{story}\), and \(N_{grp}\) are the total number of structural members, link web stiffeners, stories, and member groups in the EBF, respectively; \(A_{i}\) and \(L_{i}\) are the cross-sectional area and length of the structural member, respectively; \(V_{stf,j}\) is the stiffener volume; \(w\) is the material's unit density; \(\phi\) is the resistance factor; \(R_{u}\) and \(R_{n}\) are the demand and capacity, respectively, in terms of the generic strength, including the axial force, shear force, and moments; \(b/t\) is the width-to-thickness ratio; \(\theta_{p}\) is the link plastic rotation, which can be estimated using Equation (7), where \(\theta_{pa}\) is the allowable value; \(d_{c,i}\) indicates the column member dimensions on the i-th story; \(\Delta_{d}/h\) is the design inter-story drift evaluated using the inter-story displacement from elastic analysis \(\Delta_{e}\), the displacement amplification factor \(C_{d}\), and the importance factor \(I_{E}\) of the building; finally, \(\Delta_{a}/h\) is the allowable inter-story drift specified in the design code. In the design of the EBF system, the link members are first determined based on elastic analysis results. Next, the design forces (i.e., demand) of the non-dissipative members are determined by incorporating the capacity-limited seismic load effect, which corresponds to the member forces of the non-dissipative members when the links have reached their ultimate strength (fully yielded and strain-hardened). Nevertheless, the member forces resulting from the inelastic behavior cannot be directly obtained via elastic analysis. Therefore, an additional analysis representing the post-yielding state of the structure, as illustrated in Figure 10, was performed to obtain the member forces due to the capacity-limited seismic load effect [16]. In the model depicted in Figure 10, the link elements were cut at their center, and roller supports were placed on the outer sides of the frame to prevent side sway.