## 3 Numerical Examples
For Case-I, since the center of story mass and the supported mass do not vary per story, there is no variation of \(\gamma\) along the height. Frame A is taken as a reference frame, and the relative stiffness values for frames B, C, and D are assigned arbitrarily (Table 3). The relative stiffness value for frame E is calculated using Equation (5). Correspondingly, the moment of inertia reduction factors, \(\alpha_{b}\) and \(\alpha_{c}\), for the frames used are shown in Table 4. From Step 9, the new moment of inertia reduction factors are 0.44 times the initial moment of inertia reduction factors. CP performance level governs for this case. For Case-II, the relative stiffness values are varied following center of supported mass variation along the height to yield \(e_{x,k}\) equal to zero. Frame C is chosen as the reference frame (Table 5). The choice of the reference frame is arbitrary as long as the frame rises to the top-story level. The reference frame C is assigned \(\alpha_{b}\) equal to 0.05 and 0.1 for the top ten stories and bottom ten stories, respectively. \(\alpha_{c}\) is set as 0.3. Since some frames have fewer columns and beams than others in a given story, modification per Step 2 is necessary (Table 6). \(\alpha_{b}\) and \(\alpha_{c}\) for frames other than the reference frame are multiples of those \(\alpha_{b}\) and \(\alpha_{c}\) values for frame C times the factors of the frames given in Tables 5 and 6. From Step 7, the new moment of inertia reduction factors, \(\alpha^{\prime}_{b}\) and \(\alpha^{\prime}_{c}\), are 0.432 times the initial moment of inertia reduction factors. LS performance level governs for this case. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Story** & **Structural Members** & **Frame A** & **Frame C (or B)** & **Frame D** & **Frame E** \\ \hline \multirow{2}{*}{1–5} & Beams & 1 & 1 & 1.51 & 1.51 \\  & Columns & 1 & 1 & 1.33 & 1.33 \\ \hline \multirow{2}{*}{6–13} & Beams & 2.46 & 1 & 1.46 & - \\  & Columns & 2 & 1 & 1.5 & - \\ \hline \multirow{2}{*}{14–20} & Beams & 1.63 & 1 & - & - \\  & Columns & 1.5 & 1 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Modification factors, \(\eta_{b}\) and \(\eta_{c}\) to account for the difference in material properties, member dimensions and the number of members for Case-II frames. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reduction Factors** & **Frame A** & **Frame B** & **Frame C** & **Frame D** & **Frame E** \\ \hline \(\alpha_{b}\) for 1 st–10 th story & 0.1 & 0.08 & 0.07 & 0.08 & \(\approx\)0.1 \\ \(\alpha_{b}\) for 11 th–20 th story & 0.05 & 0.04 & 0.035 & 0.04 & \(\approx\)0.05 \\ \(\alpha_{c}\) for 1 st story & 0.3 & 0.24 & 0.21 & 0.24 & \(\approx\)0.3 \\ \(\alpha_{c}\) for 2 nd to 20 th story & 1 & 0.8 & 0.7 & 0.8 & \(\approx\)1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Moment of inertia reduction factors for Case-I frames. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Story** & \(\gamma_{A}\) & \(\gamma_{B}\) & \(\gamma_{C}\) & \(\gamma_{D}\) & \(\alpha^{\prime}_{E}\) \\ \hline
1–20 & 1 & 0.8 & 0.7 & 0.8 & \(\approx\)1 \\ \hline \hline \end{tabular} \({}^{a}\)\(\gamma_{E}\) is calculated using Equation (5) by setting \(e_{x,k}\) to zero.  
\end{table} 
Table 3: Case-I’s relative stiffness values. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Story** & \(\gamma_{A}\) & \(\gamma_{B}\) & \(\gamma_{C}\) & \(\gamma_{D}\) & \(\alpha^{\prime}_{E}\) \\ \hline
1–5 & 1.2 & 1.1 & 1 & 1.1 & \({}^{a}\) 0.03 \\
6–9 & \({}^{a}\) 1.38 & 1 & 1 & 1.1 & - \\
10–13 & \({}^{a}\) 1.81 & 0.8 & 1 & 1.1 & - \\
14–20 & \({}^{a}\) 0.61 & 0.8 & 1 & - & - \\ \hline \hline \end{tabular} \({}^{a}\) Frame stiffnesses calculated using Equation (5) by setting \(e_{x,k}\) to zero.  
\end{table} 
Table 5: Case-II’s relative stiffness values.