## 3. Numerical Examples
### Results and Discussion  
In this paper, with respect to the simplified method, the first-mode response; first-mode and second-mode responses; and first-mode, second-mode, and third-mode responses are considered for a three-story frame, a six-story frame, and a twelve-story frame with dampers, respectively. In Equation (1), the second and third modal elastic spectral displacements and the participation functions are estimated by using the \(j\)th modal period, \(j\)th mode vector, and \(j\)th modal damping ratio, respectively, as obtained from EVA of the steel frame with dampers. To confirm the accuracy of the method used to generate an equivalent inelastic SDOF system, the periods obtained from the EVA of the elaborate analytical model are compared with the periods obtained from the EVA of the equivalent SDOF system. According to the comparison results, which are not shown in this paper, the periods of the equivalent SDOF system with multi-springs are equivalent to the periods of an elaborate analytical model. The accuracy of the simplified method is expressed by the following: (i) its bias denoted as \(a\) that is calculated by the median of the ratio of the peak inter-story drift ratio \(\theta_{i}^{P}\), which is estimated by the simplified method, with respect to the corresponding peak inter-story drift ratio \(\theta_{i}\), which is computed via NTHA using the elaborate analytical model; and (ii) its dispersion denoted as \(\sigma\) that is calculated by the standard deviation of the natural logarithms of \(\theta_{i}^{P}/\theta_{i}\). The parameters bias and the dispersion are equivalently obtained by performing a one-parameter log-log linear least-squares regression of \(\theta_{i}\) on \(\theta_{i}^{P}\). A bias exceeding unity implies overestimation, while a bias less than unity implies an underestimation of the average by the simplified method. In the paper, the accuracy of the simplified method is confirmed by using 876 results of NTHA using the elaborate analytical model. The structural demand parameter for evaluation is denoted by the maximum peak inter-story drift ratio \(\theta_{\text{max}}\), which corresponds to a maximum response over time and a peak with respect to the height of the structure, because \(\theta_{\text{max}}\) correlates well with the structural and nonstructural damage in the structure [30]. Figures 5-7 illustrate the regressions of \(\theta_{\text{max}}\) computed via NTHA using the elaborate analytical model on the \(\theta_{\text{max}}^{P}\) estimated by the simplified method for a three-story frame, a six-story frame, and a twelve-story frame with dampers, respectively, that are subject to all the earthquake records. The horizontal axis represents the maximum peak inter-story drift ratio that is computed via NTHA using the elaborate analytical model, and the vertical axis corresponds to that estimated by the simplified method. The figures also present the values of bias (\(a\)) and dispersion (\(\sigma\)), for which the solid lines and the dotted lines denote the regression lines and the lines of \(\theta_{i}^{P}/\theta_{i}=1\), respectively. Based on Figures 5-7, the maximum peak inter-story drift ratios estimated by the simplified method agree fairly well with those computed via NTHA using the elaborate analytical model, irrespective of the difference in the damper parameters with respect to the stiffness ratio \(\kappa\) and the drift ratio \(\upsilon\). The biases and dispersions of the simplified method are in the range of 0.995-1.01 and 0.041-0.053, respectively, for the three-story frame, as shown in Figure 5; 0.992-1.012 and 0.051-0.075, respectively, for the six-story frame, as shown in Figure 6; and 1.005-1.033 and 0.059-0.075, respectively, for the twelve-story frame, as shown in Figure 7. The ranges of the bias and dispersion of the simplified method increase as the height of the structure increases. However, the value does not affect the estimation of the accuracy of the simplified method because the increase in the value is very small.