## 3 Building's Modeling  
A finite element method (FEM) was constructed to model the structural system and mass distribution. Non-linear dynamic time history analyses were performed to account for geometrical and structural non-linearities. The beams and the columns were modeled as frame elements with rectangular cross sections (see Table 1), while the walls were modeled as shell elements. The rigid floor diaphragm assumption was used for the modeling of the stories, as the buildings are regular (EN 1998-1:2004, page 42, section 4.2.3.2) and in elevation (EN 1998-1:2004, page 43, section 4.2.3.3) [31]. For the walls and floors, a four-node shell elements was used in this study. The selected shells elements are homogeneous, with 6 DOF in each node, and an appropriately selected mesh was used in order to have equilibrium between the accuracy of the results and the computation cost (103,200 DOF for the \(\ll\) low-rise \(\gg\) building, 272,400 for the \(\ll\) mid-rise \(\gg\), and 628,800 for the \(\ll\) high-rise \(\gg\)). In the three-dimensional structural model, elastic flexural stiffness and shear stiffness were taken into account, and equal to the one-half of the corresponding stiffness of the uncracked \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Building Type** & **Low-Rise** & **Mid-Rise** & **High-Rise** \\ \hline Number of stories & G + 03 & G + 08 & G + 15 \\ Story height & & 3.5 m & 56 m \\ Total height including roof level & 14 m & 31.5 m & 56 m \\ Columns cross section & 30 \(\times\) 50 cm & 30 \(\times\) 80 cm & 30 \(\times\) 80 cm \\ Beams cross section & & 30 \(\times\) 30 cm & \\ Walls thickness & & 30 cm & \\ Slab thickness & & 20 cm & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the investigated buildings. Figure 1: Plan view of the buildings (units in m). elements [31]. Material properties like concrete and steel rebars remain the same for all the stories, while the building is subjected to gravity and lateral loads (see Figure 1). The optimal design of the strengthening solutions is presented in the Section 4 of this paper. A tuned-mass damper (TMD), also known as a pendulum damper, is not actually a damper, but rather a pendulum or another gravity-based oscillator that is attached to the structure in such a way that it counteracts the vibration of one or more fundamental modes, thereby reducing the wind and/or seismic response of those modes. A TMD was modeled using a spring-mass system with damping. A linear link element reproduced the spring properties, while the mass and weight was also assigned in the model. The details of the TMD mass (or PTMD) for each building are presented in the Section 4.1. The damping properties of nonlinear viscous dampers (VD) were based on the Maxwell model of viscoelasticity [32]. The nonlinear properties, as stiffness, damping coefficient, and damping exponent were specified, and modeled in series. A linear link object is most suitable unless nonlinear damping is assigned using a damping exponent other than 1.0. This enables the modeling of a linear dashpot parallel with linear stiffness for both linear and nonlinear analysis cases. The numerical modeling of friction dampers (FD) was very easy, since the hysteretic loop of the friction dampers is perfectly rectangular, similar to the perfectly elasto-plastic material. The friction dampers were modeled as a fictitious plasticity element having a yield force equal to the slip load. The FD and VD were positioned in steel diagonal brace elements. More details for the shape and the position of the steel diagonal braces are given in the Section 4 of this paper. The braces were modeled as a frame element. All building models, i.e., the benchmark buildings and the alternative ones with the passive energy dissipation systems, were analyzed for seven different real and artificial accelerograms that were compatible to ground type B-dependent Eurocode 8 elastic spectra (seismic zone V according to the French national annex [33]).