# Seismic Upgrade of an Existing Reinforced Concrete Building Using Steel Plate Shear Walls (SPSW)
## 4 Modeling of the SPSWs
### Equivalent Brace Model (EBM)  
Each SPSW is modeled by a diagonal tension-only brace, designed according to [17]. The mechanics behind this model are that a SPSW behaves like a tension-only brace, assuming the inclination angle of the tension field to be \(\alpha=45^{\circ}\). The thickness of the steel plate is determined according to the following equation [17]  
\[\mathrm{t_{w}}\geq\frac{\mathrm{V_{u}}}{\mathrm{0.90\cdot 0.42\cdot f_{y}\cdot L _{cf}\cdot sin2\alpha}} \tag{1}\]  
where L\({}_{cf}\) is the clear distance between the VBEs (columns), \(f_{y}\) is the yield stress of the steel web-plate (herein considered 235 MPa), and \(\mathrm{V_{u}}\) is the shear force taken from the modal response spectrum analysis. The thickness of the SPSW should be specified by the equation [17]  
\[\frac{\mathrm{min(L,h)}}{\mathrm{t_{w}}}\leq 25\sqrt{\frac{E}{\mathrm{F_{y}}}} \tag{2}\]  
where L and h are the length and height of the web-plate, respectively, and E is the modulus of elasticity of steel. The resulting thickness is \(\mathrm{t_{w}}=5.0\) mm for each plate. ANSI/AASC 341-22 [13] and CSA S16-19 [14] design standards require that the length-to-height (L/h) ratio of the web-plates satisfy \(0.8<L/h\leq 2.5\). Additionally, the moment of inertia \(\mathrm{I_{c}}\) of the VBEs should satisfy the following equation [13; 17]  
\[\mathrm{I_{c}}\geq\frac{\mathrm{0.00307\cdot t_{w}\cdot h^{4}}}{\mathrm{L}} \tag{3}\]  
Thus, the area A of the equivalent tension brace is calculated by [13]  
\[\mathrm{A}=\frac{\mathrm{L}\cdot t_{w}\cdot sin2\alpha}{2\cdot\Omega_{s}\cdot sin \theta} \tag{4}\]  
where \(\theta\) is the angle between the vertical and longitudinal axes of the equivalent diagonal brace, L is the distance between the centerlines of the VBEs, and \(\Omega_{s}=1.2\) is the system  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Horzontal Direction of** \\ **the Building** \\ \end{tabular} & \begin{tabular}{c} **Seismic** \\ **Motion** \\ \end{tabular} & \begin{tabular}{c} **Maximum IDR (\%)** \\ **1 st Storey** \\ \end{tabular} & \
\begin{tabular}{c} **Base Shear (kN)** \\ **2 nd Storey** \\ \end{tabular} &
\begin{tabular}{c} **Maximum RIDR (\%)** \\ \end{tabular} \\ \hline \multirow{8}{*}{X} & 1a & 1.15 & 0.51 & 0.19 & 507.3 & 0.69 \\  & 1b & 2.03 & 1.06 & 0.33 & 557.6 & 0.70 \\  & 2a & 1.19 & 0.58 & 0.17 & 464.6 & 0.61 \\  & 2b & 1.79 & 0.52 & 0.18 & 471.6 & 1.37 \\  & 3a & 0.90 & 0.53 & 0.21 & 422.1 & 0.37 \\  & 3b & 1.08 & 0.53 & 0.22 & 580.4 & 0.26 \\ \hline \multirow{8}{*}{\(\Upsilon\)} & 1a & 1.81 & 1.49 & 0.37 & 490.4 & 0.70 \\  & 1b & 0.77 & 0.53 & 0.29 & 440.9 & 0.17 \\ \cline{1-1}  & 2a & 1.15 & 0.87 & 0.26 & 441.4 & 0.89 \\ \cline{1-1}  & 2b & 1.67 & 0.63 & 0.21 & 450.1 & 1.21 \\ \cline{1-1}  & 3a & 1.09 & 0.70 & 0.26 & 410.3 & 0.21 \\ \cline{1-1}  & 3b & 1.10 & 0.63 & 0.24 & 448.8 & 0.64 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Maximum values for IDR, RIDR, and base shear.  
overstrength factor. The resulting brace sections are assumed to be of CHS type, and the resulting sections are CHS 273/16 for the first storey and CHS 244.5/16 for the second and third storeys. The model of the r/c building with SPSWs modeled by equivalent braces is shown in Figure 7. The fundamental period of the r/c building in Figure 7 is T1 = 0.739 s.