## 2 Materials and Methods  
### Case Study  
The case study has been formulated following the Ecuadorian Construction Standard (NEC-2015 [24]) guidelines to obtain the design spectrum and analyze the 2D frame system. Table 1 summarizes the data for the study. Construction of the Design Spectrum for the Case Study  
The acceleration spectrum consists of an elastic and an inelastic configuration. The latter is reduced with the corresponding response coefficient. The elastic response spectrum of accelerations \(S_{a}\) is expressed as a fraction of the acceleration of gravity, for the design \begin{table}
\begin{tabular}{c c l} \hline \hline  & & **Date** \\ \hline \(V\) & & Seismic zone \\ Z & 0.4 & Seismic zone factor \\ \(\Phi_{P}\) & 1 & Irregularity Factor in Plant \\ \(\Phi_{e}\) & 1 & Irregularity Factor in Elevation \\ \(R\) & 8 & Structural Response Reduction Coefficient \\ \(I\) & 1 & Importance Factor \\ Soil & D & Soil profile type \\ \hline \hline \multicolumn{3}{c}{**Factors**} \\ \hline \(F_{a}\) & 1.2 & Soil amplification coefficient in the short-period zone \\ \(F_{d}\) & 1.19 & Displacement for design in rock \\ \(F_{s}\) & 1.28 & Nonlinear behavior of soils \\ \(\eta\) & 2.48 & Ratio between spectral acceleration \(S_{a}\) and \(PGA\) \\ \(r\) & 1 & Factor by geographic location \\ \(T\) & 0.662 & Structure vibration period \\ \(T_{0}\) & 0.127 & Lower limit period of vibration in the elastic spectrum of accelerations \\ \(T_{C}\) & 0.698 & Upper limit period of vibration in the elastic spectrum of accelerations \\ \(C_{t}\) & 0.072 & Coefficient that depends on the type of building \\ \(\alpha\) & 0.8 & Semi-space impedance \\ \(h_{n}\) & 16 & Total height of the building [m] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters to obtain the design spectrum, according to NEC-2015. earthquake levels. Equations (1)-(3) represent the slopes of the elastic response spectrum (blue curve in Figure 1) only for vibration modes other than the fundamental one. \[T<T_{0} S_{a1}(T)=Z\,F_{a}\bigg{(}1+(\eta-1)\frac{T}{T_{0}}\bigg{)} \tag{1}\]  
\[T_{0}<T\leq T_{\mathrm{C}} S_{a2}(T)=\eta\,Z\,F_{a} \tag{2}\]  
\[T>T_{\mathrm{C}} S_{a3}(T)=\eta\,Z\,F_{a}\bigg{(}\frac{T_{\mathrm{C}}}{T}\bigg{)}^{r} \tag{3}\]  
\[S_{a}(T)(g), S_{a}(T)\frac{1}{R\,\Phi_{p}\,\Phi_{e}}(g) \tag{4}\]  
The forces obtained through dynamic elastic analysis can be reduced for design purposes by dividing by the corrected R-value and multiplying by importance factor I (in expression (4): left factor for the curve in blue, right factor for the curve in red). Therefore, a spectrum of reduced accelerations can be considered (red curve in Figure 1). The above graph and expressions have been obtained directly from the output of MATHCAD 7.00 software (trial version), under regulation 2-NEC-SE-DS-Seismic-Hazard-part-1, article 3.3.1: Spectrum horizontal elastic design in accelerations. For the structural system, the main geometries have been considered, with the formulations below, used to deduce the static basal shear. Once the vibration periods of the structure are obtained, it is done the calculation of spectral accelerations. The case of the study considered holds that \(T_{0}<T\leq T_{\mathrm{C}}\) and the static basal shear is determined for the direction in \(X\), by the following equation:  
\[C=\frac{I\,S_{a2}}{R\,\Phi_{p}\,\Phi_{e}}=0.149\]  
The geometry of the 2D portico is presented below and consists of 5 floors, three sections, a length between sections of 6.40 m (Length), and a floor height of 3.20 m (Height) (Figure 2). Figure 1: Elastic response spectrum of accelerations (curve in blue) and inelastic response spectrum of accelerations (curve in red). For the analytical calculation process, \(HEB450\) columns and \(1 PE400\) beams were assigned to the 2D frame (Figure 2). The following loading system was applied:  
\[CP=24\frac{\text{kN}}{\text{m}}:\ Permanent\ load.\]  
\[CV=12\frac{\text{kN}}{\text{m}}:\ Variable\ load.\]  
\(SH\): Horizontal Quake. \(NL_{X}CP\): Notional Load of permanent load in \(X\) direction.