## 3 Numerical Examples
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Result** & **Earthquake Name** & **Station Name** & **Magnitude** & **Mechanism** & **Peak Ground** & **Scale** & **Tp-Pulse** & **Arias Intensity** & **V530** & **5-95\%** \\
**ID** & **The** & **In** & **In** & **In** & **In** & **In** & **In** & **In** & **In** & **In** \\
2 & **“The** & **“The** & **“The** & **In** & **In** & **In** & **In** & **In** & **In** & **In** & **In** \\
3 & **“Cape Mendeno** & **“Fortuna-Fortuna Blvd** & 7.01 & & & & & & & & & \\
4 & **“Landes** & **“Nern-Parting Blvd** & 7.28 & & & & & & & & \\
5 & **“Kocaeli, Turkey** & **“Arcella** & 7.51 & & & & & & & & \\
6 & **“Kocaeli, Turkey** & **“Duczer** & 7.51 & & & & & & & & \\
7 & **“Kocaeli, Turkey** & **“Cebrav** & 7.51 & & & & & & & & \\
8 & **“Chi-Chi, Taiwan** & **“CHV006** & 7.62 & & & & & & & & \\
9 & **“Chi-Chi, Taiwan** & **“CHV010** & **7.62** & & & & & & & & \\
10 & **“Chi-Chi, Taiwan** & **“CHV024** & 7.62 & & & & & & & & \\
11 & **“Chi-Chi, Taiwan** & **“CHV029** & **7.62** & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected ground motions for Design Basis Earthquake (DBE). To define the secant stiffness of the case studies, the moment of inertia reduction factors, \(\alpha_{b}\) for beams and \(\alpha_{c}\) for first-story columns, of the reference frames need to be assumed. [PERSON] (1966) and [PERSON] et al. [PERSON] et al. (1966) adopted 10 percent of the uncracked section stiffness to represent the secant stiffness of RC bridge piers as a start for their iterative design methods. [PERSON] (1966) determined the member stiffness matrix with plastic hinges at one or both ends. Accordingly, for members with hinges at both ends, the stiffness of that member will be the stiffness based on the gross moment of inertia of members times the ratio of post-yield stiffness to the elastic stiffness of the plastic hinge. For members with only one hinge, no such simplification has been made. Thus, if that ratio is assumed as 0.1 for members with two hinges at member ends, the moment of inertia reduction factor, \(\alpha_{b}\), will be also 0.1. The values of the reduction factors are chosen arbitrarily as a starting value as it does not affect the values for \(\alpha^{\prime}_{c}\) and \(\alpha^{\prime}_{b}\). Therefore, \(\alpha_{c}\) is assigned 0.3 since only one plastic hinge is expected and \(\alpha_{b}\) as 0.1 or, in some cases, 0.05. For the reference frames of the case studies, \(\alpha_{c}\) is set as 0.3, whereas \(\alpha_{b}\) for the lower ten story beams is 0.1 and for the top 10 stories 0.05. A lower value of \(\alpha_{b}\) at the top stories is chosen because seismic demand reduces as the height increases. Steps 3 to 5 need to be performed for both LS and CP levels. When using Step 5, the peak value of the mean displacement response spectra for the equivalent viscous damping may be less than the initial design displacement at effective height, \(\Delta_{d,initial}\). In that case, a method discussed in Section 3.4.6(b) of [PERSON] et al. [PERSON] et al. (1966) is employed. Hence, lower final displacement, \(\Delta_{d,final}\), at reduced equivalent damping ratio is found. To attain \(\Delta_{d,final}\), the design story displacements need to be factored by the ratio \(\Delta_{d,final}/\Delta_{d,initial}\). The DDBD base shear, \(V_{b}\), for the two performance levels is compared, and the maximum is taken as a governing design base shear. When the value of P-\(\Delta\) force at story \(i\), \(Q_{i}\) is found to be negative, it is not added to the lateral force \(F_{i}\) and, hence, only \(F_{i}\) is taken as the design force at that story. Column design moments, \(M_{D}\), are amplified by the ratio \(\varphi^{o}/\varphi_{s}=1.2\) using \(\varphi^{o}=1.1\) and \(\varphi_{s}=0.9\). Figure 6: (**a**) Acceleration response spectrum for DBE. (**b**) Acceleration response spectrum for Maximum Considered Earthquake (MCE).