# Experimental Evaluation of Rigidity Center
## 3 Results and Discussion
### Interpreting the Results and Details of the Comparisons
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Lead disc**} & **Membour** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **d\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** & **b\({}_{w}\), max** \\ \hline \multirow{6}{*}{**S**} & S01 & 3 & 100 & 250,000 & 962 & 100 & 26.5 & 1.25 & 20,30,122 & 35,50,48 & 94,508 & \multirow{2}{*}{**R**} & 27.24 \\  & C24 & 6 & 6 & 108 & 1977 & 100 & 272 & 21.65 & 2.48 & 53.79 & 14,031 & (from each side) \\  & C24 & 6 & 6 & 108 & 1977 & 100 & 28 & 1.38 & 2.48 & 3.42 & 96 & \\  & C24 & 6 & 6 & 108 & 1977 & 100 & 272 & 21.65 & 2.48 & 53.79 & 14,483 & \\ \hline \multirow{6}{*}{**S**} & S01 & 100 & 3 & 225 & 962 & 100 & 225 & 6.40 & 25.55 & 163.57 & 36,803 & \multirow{2}{*}{**R**} & **26.52** \\  & C24 & 6 & 6 & 108 & 1977 & 100 & 272 & 6.08 & 2.48 & 151.1 & 4199 & 193.52 \\ \cline{1-1}  & C24 & 6 & 6 & 108 & 1977 & 100 & 28 & 7.75 & 2.48 & 19.25 & 559 & \\ \cline{1-1}  & C24 & 6 & 6 & 108 & 1977 & 100 & 28 & 7.75 & 2.48 & 19.25 & 559 & \\ \hline \end{tabular}
\end{table} 
Table 5: Rigidity center calculation of A2. * The meshing information of the area elements (for fast solutions, sometimes shear walls are not meshed);
* The missing slab information (for fast solutions, sometimes slabs are not used in the model; instead, their rigid diaphragm properties are only used). Due to the abovementioned information, the approaches do not calculate the rigidity centers close to each other in asymmetric buildings. The global rigidity center in the floor plane will not be correct; instead, a plasmatic axis in 3D is usually recommended [41; 42; 43]. As significant differences exist, their reliability was questioned by the authors, and analytical studies were conducted to determine their reliability on full-scale buildings. The results indicated that the third approach given above may be the most reliable one [44]. After determining the rigidity center locations experimentally, all the buildings were 3D-modeled using ETABS. During the 3D modeling of the buildings using ETABS, all the columns were modeled using frame elements with the same size of balsa sticks, and the material properties were determined experimentally considering each column individually. A thin shell was used to model the shear walls, foundation and slabs with the same size and material properties used in the experiments. Uncracked sectional properties were utilized, and dynamic analysis was performed taking only self-weight into account. The rigidity center locations were found using dynamic analysis to compare the test results with the analytical solutions. In ETABS, a unit horizontal load in \(x\) and \(y\) directions and a unit moment in \(z\) direction were applied to any point on the floor level (Figure 13), and the RC was calculated by proportioning the rotations obtained as a result of these loadings (Equation (5)). In the equation, \(x_{r}\) and \(y_{r}\) show the RC location in \(x\) and \(y\) directions, respectively, from the reference axis; and \(R_{zx}\), \(R_{zy}\) and \(R_{zz}\) are the rotations of the relevant node in \(x\), \(y\) and \(z\) directions, respectively [23]:  
\[x_{r} =\frac{-R_{zy}}{R_{zz}},\ y_{r} =\frac{R_{zx}}{R_{zz}} \tag{5}\]  
The RCs calculated from the test results were marked on each floor, as shown in Figure 13, and they were compared to the ones obtained from the 3D modeling in ETABS (E). In addition to the RCs, the mass center (MC) was also marked on the floor plans. In Table 6, the \(x\) and \(y\) distances from the reference axes (\(x_{r}\), \(y_{r}\)) were given comparatively, and the difference between the results was calculated according to Equation (6) [45]. It can be seen for A2 that E and RC are almost the same, and the maximum difference is 1.44%, meaning that E can predict the rigidity center accurately. Figure 13: Rigidity center procedure in ETABS.