## 3 Application of Theoretical Framework  
### Data Generation  
In this paper, several steel shear-buildings discretized as MDOF structures with different structural vibration period T\({}_{0}\) were analyzed using wind loads from synthetic records of 10 min. The case studies are defined by the combinations of some parameters that consider diverse wind conditions and structural characteristics (see Figure 5). The range of values for each parameter was selected to cover a considerable number of sensitive structures to the dynamic effects of the wind. The values of basic wind speed or regional speed are presented in 3-s gust wind speeds because this is more common in this type of design; however, the time scale factor was considered for 10 min. The slenderness relation is defined as \(\mathrm{R}_{e}=\mathrm{H}/\mathrm{b}\), where H is the total height of the structure and b is the plane dimension parallel to the direction of the wind (see Figure 4a). The stiffness ratio corresponds to the reduction of the stiffness along the height between the first and the last story level, i.e., \(\mathrm{R}_{k}=k_{\mathrm{N}_{m}}/k_{1}\) (see Figure 4b). The distributed mass represents the mass concentration per unit area for each floor, and this parameter defines the lumped mass m of each story level. The dynamic loads are generated from synthetic records, where the turbulent parts of the longitudinal components are obtained by the Shinozuka method. According to [PERSON], the wind is air mass with kinetic energy. It can be expressed in terms of pressure or force acting on a contact area; the mathematical form for a structure is [31]:  
\[\mathrm{F}_{j}(t)=\frac{1}{2}C_{d}\rho A_{j}\Big{(}\mathrm{U}_{j}(t)-\dot{d}_{ j}(t)\Big{)}^{2} \tag{50}\]  
where \(\mathrm{F}_{j}(t)\) is the wind force on the \(j\)-th floor; \(C_{d}\) is the drag coefficient, for rectangular prismatic \(C_{d}\approx 1\)[31], which is used to quantify the drag or resistance of an object in a fluid environment; \(\rho\) is the density of the air, for this study \(\rho=1.25\) kg/m\({}^{3}\); \(A_{j}\) is the contact area on the \(j\)-th floor; and \begin{table}
\begin{tabular}{l} \hline \hline
**Step-by-step procedure:** \\ \hline Initial calculations: \\ \(1.1\ \bar{\mathbf{K}}=\overline{\mathbf{K}}+\frac{\gamma}{p\Delta t}\overline{ \mathbf{C}}+\frac{1}{\beta(\Delta t)^{\mathrm{T}}}\overline{\mathbf{M}}\) \\ \(1.2\ \mathrm{a}=\frac{1}{p\Delta t}\overline{\mathbf{M}}+\frac{\gamma}{2} \overline{\mathbf{C}},\mathrm{b}=\frac{1}{2p}\overline{\mathbf{M}}+\Delta t \Big{(}\frac{\gamma}{2p}-1\Big{)}\overline{\mathbf{C}}\) \\ Calculations for each time step \(i\) \\ \(2.1\ \bar{\mathbf{F}}_{i}=\mathbf{\Phi}^{\mathrm{T}}\mathbf{F}_{i}\) \\ \(2.2\ \Delta\bar{\mathbf{F}}_{i}=\mathbf{\bar{A}}\bar{\mathbf{F}}_{i}+\mathrm{a} \dot{\mathbf{q}}_{i}+\mathrm{b}\ddot{\mathbf{q}}_{i}\) \\ \(2.3\ \mathbf{a}\mathbf{q}_{i}=\mathbf{\bar{K}}^{-1}\mathbf{\bar{A}}\bar{\mathbf{F}}_{i}\) \\ \(2.4\ \Delta\dot{\mathbf{q}}_{i}=\frac{p\chi}{p\Delta t}\Delta\mathbf{q}_{i}-\frac{ \gamma}{2p}\dot{\mathbf{q}}_{i}+\Delta t\Big{(}1-\frac{\gamma}{2p}\Big{)} \ddot{\mathbf{q}}_{i}\) \\ \(2.5\ \Delta\ddot{\mathbf{q}}_{i}=\frac{1}{p\Delta t}\Delta\mathbf{q}_{i}-\frac{1}{p \Delta t}\dot{\mathbf{q}}_{i}-\frac{1}{2p}\ddot{\mathbf{q}}_{i}\) \\ \(2.6\ \mathbf{q}_{i+1}=\mathbf{q}_{i}+\Delta\mathbf{q}_{i}\dot{\mathbf{q}}_{i+1}= \dot{\mathbf{q}}_{i}+\Delta\dot{\mathbf{q}}_{i},\ddot{\mathbf{q}}_{i+1}=\ddot{ \mathbf{q}}_{i}+\Delta\ddot{\mathbf{q}}_{i}\) \\ \(2.7\ d_{i+1}=\Phi\mathbf{q}_{i+1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Procedure of the Newmark method for multi-degrees of freedom systems. \(\mathrm{U}_{j}(t)\) is the longitudinal component of the wind. It is defined as the sum of the mean velocity and its corresponding turbulent part \(\mathrm{U}_{j}(t)=\overline{\mathrm{U}}_{j}+u_{j}(t)\); \(\dot{d}_{j}(t)\) is the velocity of the mass lumped on the \(j\)th floor.