## Theoretical Framework
### Dynamic Response  
The Newmark method is one of the numerical integration methods most frequently used in structural dynamic analysis; software packages such as OpenSees (v2002, CA, USA), Ruaumoko (v2004, Christchurch, New Zealand) and SAP2000 (v2012, CA, USA) [27; 28; 29] include routines of dynamic analysis by this method. This procedure is based on two types of assumptions in order to simplify and solve the differential equations of motion. The assumptions are focused on determining the acceleration between two time instants, which may be linear or a constant mean value. The idealization of a prismatic building as shear-frame or shear-building (Figure 4) can be represented by a Multi-Degree-of-Freedom (MDOF) system [30]. The governing equation of motion for MDOF systems subjected to external forces can be given in a matrix form as next:  
\[\mathbf{M}\ddot{\mathbf{d}}+\mathbf{C}\dot{\mathbf{d}}+\mathbf{K}\mathbf{d}=\mathbf{F}, \tag{43}\]  
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices, respectively, \(\mathbf{d}\), \(\dot{\mathbf{d}}\) and \(\ddot{\mathbf{d}}\) are the displacement, velocity and acceleration vectors; and \(\mathbf{F}\) is a matrix with the force vectors. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Terrain Category** & \(\mathbf{z_{0}}\)** **(m)** & \(\overline{\mathbf{b}}\) \\ \hline I—Lakes or flat and horizontal area with negligible vegetation and without obstacles & 0.01 & 1.17 \\ \hline II—Area with low vegetation such as grass and isolated obstacles (trees, buildings) with & 0.05 & 1.00 \\ separations of at least 20 obstacle heights & & \\ \hline III—Area with regular cover of vegetation or buildings or with isolated obstacles with & & \\ separations of maximum 20 obstacle heights (such as villages, suburban terrain, & 0.30 & 0.77 \\ permanent forest) & & \\ \hline IV—Area in which at least 15\% of the surface is covered with buildings and their average & 1.00 & 0.55 \\ height exceeds 15 m & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Terrain categories and terrain parameters based in Eurocode [22]. The matrix form of the mass and stiffness matrices for a MDOF structure (Figure 4b) with shear resistant are:  
\[\mathbf{M}=\left[ \begin{array}{cccccc}\text{m}&0&\cdots&\cdots&0\\ 0&\text{m}&0&\cdots&0\\ \cdots&0&\cdots&0&\cdots\\ 0&\cdots&0&\text{m}&0\\ 0&\cdots&\cdots&0&\text{m}\end{array} \right];\mathbf{K}=\left[ \begin{array}{ ccccc}\text{k}_{1}+\text{k}_{2}&-\text{k}_{2}&\cdots&\cdots&0\\ -\text{k}_{2}&\text{k}_{2}+\text{k}_{3}&\cdots&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&\cdots&\cdots&\text{k}_{N_{m-1}}-\text{k}_{N_{m}}&-\text{k}_{N_{m}}\\ 0&\cdots&\cdots&-\text{k}_{N_{m}}&\text{k}_{N_{m}}\end{array} \right] \tag{44}\]  
\(\ldots\)  
The form of the damping matrix \(\mathbf{C}\) depends on the damping characteristic of the structure, which can be classical or nonclassical damping. The classical damping is an appropriate idealization if similar damping mechanisms are distributed throughout the structure, e.g., a multistory building with a similar structural system and structural materials over its height [30]. In this work, the Rayleigh Damping method is used in order to build a classical matrix \(\mathbf{C}\). The damping matrix can be approached by the following equation of Rayleigh:  
\[\mathbf{c}=a_{0}\mathbf{M}+a_{1}\mathbf{K}, \tag{45}\]  
where \(a_{0}\) and \(a_{1}\) Rayleigh coefficient and could be estimated by:  
\[a_{0}=\zeta\frac{2\omega_{i}\omega_{j}}{\omega_{i}+\omega_{j}};\;a_{1}=\zeta \frac{2}{\omega_{i}+\omega_{j}}, \tag{46}\]  
where \(\zeta\) is the damping ratio for the first mode; and \(\omega_{i}\) and \(\omega_{j}\) are two angular frequencies that correspond to two adjacent vibration modes.