## 3 Case Study
However, by imposing a diaphragm constraint at each of the 10 stories of the structure, the number of displacement components reduces to 390, corresponding to the translations \(TX\), \(TY\), and rotation \(RZ\) computed in the mass centers of the slabs, and the remaining rotations \(RX\), \(RY\) and translation \(TZ\) computed at \begin{table}
\begin{tabular}{c c} \hline \hline
**Memorized Variable** & **Role in Algorithm** \\ \hline \(\theta^{\prime}\)-the plastic rotation & Used to determine whether the structural \\ behavior is elastic or plastic. \\ \hline The peak positive pair \((\theta^{\prime},\mathbf{m}(\mathbf{t}))\) in the first quadrant & Used to define the reloading slopes originated in the \\ \hline The peak negative pair \((\theta^{\prime},\mathbf{m}(\mathbf{t}))\) in the third quadrant & Used to define the reloading slopes originated in the \\ \hline The current quadrant location of \((\theta^{\prime\prime},\mathbf{m}(\mathbf{t}))\) pair & Used to distinguish between \\ The previous quadrant location of \((\theta^{\prime\prime},\mathbf{m}(\mathbf{t}))\) pair & different reloading situations \\ \hline The last plastic rotation associated zero & Used to define the reloading slopes \\ bending moment & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak-oriented algorithm memororized variables. Figure 4: Moment-plastic rotation cyclic model used in this paper. each joint. The use of a diaphragm constraint is considered to be a good balance between numeric efficiency and accuracy. Additional coordinates condensation can be performed as it is suggested in [3], but the reader is suggested to carefully consider such techniques since the member restoring force matrix \(\mathbf{K_{R}}\) may become singular once a static condensation is performed. It must be mentioned that the diaphragm constraint implemented here will not produce singular member restoring force matrices \(\mathbf{K_{R}}\). As a conclusion, the numerical model used in this case study is described by 1600 unknowns (390 displacements, 390 velocities, and 820 plastic rotations) that require evaluation in each time step. Regarding the structural elements, beams bending capacity was considered 520 kN and the column yielding surface \(f(x,y,z)\) (Figure 6) was defined as previously proposed by [PERSON] et al. [30]:  
\[f(m_{y},m_{z},P)=\sqrt{\left(\frac{m_{y}}{m_{yp}}\right)^{2}+\left(\frac{m_{z }}{m_{zp}}\right)^{2}}+\left(\frac{P-P_{b}}{P_{p}}\right)^{2}-1 \tag{32}\]  
where \(m_{yp}=m_{zp}=2000\) kNm define the maximum bending moments, \(P_{p}=-11\),500 kN is the vertical semi-axis, \(P_{b}=-8500\) kN is the balance point axial force, and \((m_{y},m_{z},P)\) are the member forces varying along \(y\), \(z\), and \(x\) axis respectively. Furthermore, \(P_{b}+P_{p}\) is the cross-section compression capacity and \(P_{b}-P_{p}\) is the cross-section tension capacity. Simulation of the dynamic nonlinear structural model was carried out using the North-South and East-West Vrancea 1977 seismic recordings. It must be noted that the defining characteristic of these motions is that they exhibit pulse-like behavior, most of the seismic input energy being concentrated in a short amount of time [31]. Figure 5: The ten story 3D framed structure.