# Evaluation of Link Overstrength Factor for the Seismic Design of Eccentrically Braced Frames
## 4 Structural Analysis  
### Nonlinear Analysis Model  
In this section, nonlinear analyses are performed to ascertain whether the concept of capacity design principle is satisfied (i.e., whether the non-dissipative member yields) and to identify the link overstrength factor necessary to prevent the non-dissipative member from yielding. To conduct nonlinear analyses, the elastic analysis models obtained using the numerical procedure based on the multi-objective optimization described in the previous section were transformed into nonlinear models. Only the EBF frames were explicitly modeled, whereas the gravity frames were lumped into leaning columns to represent the P-delta effect. Nonlinear properties were modeled exclusively at the links because nonlinear behavior is expected only at these links based on the capacity design principle. Nonlinear time history analyses were conducted using OpenSees [20]. Figure 11: Optimization results (D-S15B9L1 performance group). Figure 12: Structural quantities by performance group: (**a**) five-story building; (**b**) ten-story building; (**c**) fifteen-story building. Several modeling approaches [22; 23; 24; 25; 26; 27; 28; 29; 30] have been proposed to simulate the nonlinear shear and flexural behaviors of links. In this study, a link model based on [PERSON]'s interaction model [17] was adopted. As shown in Figure 13, the link model consisted of a link beam (LB), two translational springs (TSs), and two rotational springs (RSs). The length of the LB is identical to that of the actual link and is modeled as a rigid beam element because the elastic stiffness is already accounted for in the springs. The TSs and RSs are zero-length elements that simulate nonlinear shear and flexural behavior, respectively. A bilinear material model (Steel01) in OpenSees [20] was used for the TS and RS components. Table 2 presents the nonlinear modeling properties of the spring elements. The notation used in Table 2 is also shown in Figure 14.  
The expected yield strengths of the TS and RS elements, \(V_{pe}\) and \(M_{pe}\) in Figure 14, respectively, were calculated using Equation (4b,c), respectively. In the case of short links, which are governed by shear yielding, the TS element attains its ultimate strength, including strain-hardening behavior, while the RS element remains elastic. In the case of long links, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Element**} & **Yield** & **Elastic** & **Yield** & **Ultimate Plastic** & **Ultimate** & **Strain Hardening** \\  & **Strength** & **Stiffness** & **Deformation** & **Deformation** & **Overstrength Factor** & **Ratio** \\ \hline TS & \(V_{pe}=R_{y}V_{p}\) & \(K_{v}=\frac{2G\Lambda_{v}}{e}\) & \(\delta_{y}=\frac{V_{pe}}{K_{v}}\) & \(\delta_{pu}=0.08\times\frac{\xi}{2}\) & \(\Omega_{pu}=\frac{V_{p}}{R_{y}V_{p}}=1.5\) & \(b_{v}=\frac{(\Omega_{v}-1)V_{pe}}{K_{v}\sigma_{pu}}\) \\ \hline RS & \(M_{pe}=R_{y}M_{p}\) & \(K_{m}=\frac{6 EI}{e}\) & \(\theta_{y}=\frac{M_{pe}}{K_{m}}\) & \(\theta_{pu}=0.02\) & \(\Omega_{mux}=\frac{M_{v}}{R_{y}M_{p}}=1.5\) & \(b_{m}=\frac{(\Omega_{mux}-1)M_{pe}}{K_{m}\sigma_{pu}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nonlinear properties of the spring elements. Figure 14: Load–displacement relationship of the link: (**a**) TS element; (**b**) RS element. Figure 13: Analytical model for link members. the TSs and RSs behave in a manner opposite to that observed in short links. The ultimate plastic rotations for the TSs and RSs were set as 0.08 and 0.02 rad, respectively. These values are identical to the allowable plastic rotation angles for short and long links, as specified in AISC 341 [1] (Figure 6). In the case of intermediate links, shear and moment yielding can occur simultaneously. Given that the shear and moment are in equilibrium, if yielding occurs in either of these and the strength increases owing to strain hardening, subsequent yielding occurs in the other. Consequently, the final \(\Omega\) factor at the instant of link failure becomes less than 1.5.