## 2 Materials and Methods
### Numerical Study
#### 2.3.4 Procedures and Response Variables  
For a given system \(ns\) included in case \(nc\), three different procedures are used to find the solution for each register \(r\) in the GMS. These procedures are described in Table 1, where the notation used in the study is introduced. The procedures listed in Table 1 (TH, THMA, RSA-SRSS and RSA-CQC), are described hereby:  
* Time-History (TH): the complete response \({}^{r}_{ns}\mathbf{u}(t)\), \({}^{r}_{ns}\mathbf{v}(t)\), \({}^{r}_{ns}\mathbf{a}(t)\) (relative displacement, relative velocity, and absolute acceleration, respectively) of the system defined by \({}_{ns}\mathbf{m}\), \({}_{ns}\mathbf{k}\) and \({}_{ns}\mathbf{c}\) (non-classically damped system) to each scaled register \(r\) in the GMS is calculated by the central difference step-by-step method [14]. The corresponding restoring forces \({}^{r}_{ns}\mathbf{f}_{\text{S}}(t)\) and damping forces \({}^{r}_{ns}\mathbf{f}_{\text{D}}(t)\) are obtained as \({}^{r}_{ns}\mathbf{k}\cdot^{r}_{ns}\mathbf{u}(t)\) and \({}^{r}_{ns}\mathbf{c}\cdot^{r}_{ns}\mathbf{v}(t)\), respectively. Other variables are derived from these. The peak response for each variable and story is preserved. * Time-history modal analysis (THMA): the complete response \({}^{r}_{ns}\mathbf{u}_{1}(t)\), \({}^{r}_{ns}\mathbf{v}_{1}(t)\), \({}^{r}_{ns}\mathbf{a}_{1}(t)\) of the system defined by \({}_{ns}\mathbf{m}\), \({}_{ns}\mathbf{k}\) and \({}_{ns}\mathbf{c}\)approx (approximate classically damped system) to each scaled register \(r\) in the GMS, are calculated adding the responses obtained for each mode at each instant by the central difference step-by-step method. As in the previous procedure, restoring forces \({}^{r}_{ns}\mathbf{f}_{\text{S}}(t)\) and damping forces \({}^{r}_{ns}\mathbf{f}_{\text{D}}(t)\) are obtained as \({}^{r}_{ns}\mathbf{k}\cdot^{r}_{ns}\mathbf{u}_{1}(t)\) and \({}^{r}_{ns}\mathbf{c}\cdot^{r}_{ns}\mathbf{v}_{1}(t)\), respectively; the forces obtained for every time-step are not in equilibrium, because \({}^{r}_{ns}\mathbf{f}_{\text{D}}(t)\) is calculated using \({}_{ns}\mathbf{c}\) whereas \({}^{r}_{ns}\mathbf{u}_{1}(t)\) and \({}^{r}_{ns}\mathbf{v}_{1}(t)\) are calculated using \({}_{ns}\mathbf{c}\)approx, as is usual in design applications. Peak responses for every variable and story are preserved. THMA avoids the inaccuracies involved in the use of approximate modal combination rules, therefore only the first type of error discussed in Section 2.1 (due to neglected off-diagonal terms of the modal damping matrix) is present; this procedure is identified by subscript '1'. * Response spectrum analysis (RSA-SRSS and RSA-CQC): for every mode \(j\) of system \(ns\) defined by \({}_{ns}\mathbf{m}\), \({}_{ns}\mathbf{k}\) and \({}_{ns}\mathbf{c}\)approx (approximate classically damped system), and for every register \(r\) in the GMS, the modal peak relative displacement \({}^{r}_{ns}\mathbf{u}_{2\text{peak},j}\), modal peak relative velocity \({}^{r}_{ns}\mathbf{v}_{2\text{peak},j}\), and modal peak absolute acceleration \({}^{r}_{ns}\mathbf{a}_{2\text{peak},j}\), are foundusing the piecewise step-by-step method [43]. These vectors contain the peak values for every story at mode \(j\), for example \({}_{ns}^{r}\mathbf{u}_{2,\text{peak},j}=\left[{}_{ns}^{r}\mathbf{u}_{2,\text{ peak},1,j},\dots,{}_{ns}^{r}\mathbf{u}_{2,\text{peak},N,j}\right]^{T}\), etc. The modal peak values of restoring forces \({}_{ns}^{r}\mathbf{f}_{\text{52,peak},j}\) (\(={}_{ns}^{r}\mathbf{k}\cdot_{ns}^{r}\mathbf{u}_{2,\text{peak},j}\)) and damping forces \({}_{ns}^{r}\mathbf{f}_{\text{D2,peak},j}\) (\(={}_{ns}^{r}\mathbf{\zeta}\cdot_{ns}^{r}\mathbf{v}_{2,\text{peak},j}\)) are found. The modal peak values of other variables are derived from these. The peak response for every variable and story is obtained applying the SRSS rule (RSA-SRSS), identified with subscript '2', or CQC rule (RSA-CQC), identified with subscript '3'.