## 3 Material Test and Results for LLFD
### LfD Material Test Results and Analysis  
#### 3.3.1 Comparison and Analysis of Load-Displacement Curves  
Figure 9 shows the situations for the FP-LLFD and GS-LLFD specimens at a final control displacement of 33 mm. Figure 10 compares the load-displacement curves of the specimens under a torque pressure of 100-300 N-m, while Table 4 lists the yield displacements and yield loads of the specimens. Figure 8: Repeated loading plan. Figure 7: Specim installation and measurement device (Top view). Figure 10: Load–displacement curve: (**a**) FP-LLFD-1; (**b**) FP-LLFD-2: (**c**) GS-LLFD-1; (**d**) GS-LLFD-2. Figure 9: Specimens at final control displacement of 33 mm: (**a**) FP-LLFD; (**b**) GS-LLFD. The FP-LLFD-1 specimen exhibited yield displacements and yield strengths of 0.11 mm and 22.6 kN under 100 N-m torque, 0.41 mm and 76.1 kN under 200 N-m torque, and 0.41 mm and 151.0 kN under 300 N-m torque, respectively. The FP-LLFD-2 specimen showed yield displacements and yield strengths of 0.04 mm and 23.4 kN under 100 N-m torque, 0.11 mm and 81.2 kN under 200 N-m torque, and 0.08 mm and 159.8 kN under 300 N-m torque, respectively. The GS-LLFD-1 specimen showed yield displacements and yield strengths of 0.04 mm and 20.2 kN under 100 N-m torque, 0.08 mm and 145.0 kN under 200 N-m torque, and 0.18 mm and 252.2 kN under 300 N-m torque, respectively. The GS-LLFD-2 specimen exhibited yield displacements and yield strengths of 0.36 mm and 33.2 kN under 100 N-m torque, 0.17 mm and 111.7 kN under 200 N-m torque, and 0.39 mm and 238.7 kN under 300 N-m torque, respectively. Figure 11 shows the relationships between the torque pressure and yield strength for the FP-LLFD and GS-LLFD specimens. The figure shows the theoretical design strengths calculated by Equation (1) and an approximation equation based on regression analysis for comparison with the test results. Overall, the test results for the FP-LLFD and GS-LLFD in the 100-300 N-m torque range were approximately twice as low as the design strength. Therefore, the non-linear dynamic analysis of two-story R/C frames with LLFDs described in Sections 5 and 6, as well as a non-linear dynamic analysis of the entire building for seismic retrofitting design \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Specimens**} & \multirow{2}{*}{
\begin{tabular}{c} **Torque** \\ **(N-m)** \\ \end{tabular} } & \multicolumn{2}{c}{**Yield State**} & \multicolumn{2}{c}{**Maximum Strength**} \\ \cline{3-6}  & & **Displacement** & **Strength** & **Positive** & **Negative** \\  & & \(\delta_{d,y}\) **(mm)** & \(V_{d,y}\) **(kN)** & \(V_{d,max}\) **(kN)** & \(V_{d,min}\) **(kN)** \\ \hline \multirow{3}{*}{FP-LLFD-1} & 100 & 0.11 & 22.6 & 24.7 & 23.0 \\ \cline{2-6}  & 200 & 0.41 & 76.1 & 86.8 & 85.0 \\ \cline{2-6}  & 300 & 0.41 & 151.0 & 192.0 & 190.3 \\ \hline \multirow{3}{*}{FP-LLFD-2} & 100 & 0.04 & 23.4 & 29.8 & 27.0 \\ \cline{2-6}  & 200 & 0.11 & 81.2 & 119.6 & 116.0 \\ \cline{2-6}  & 300 & 0.08 & 159.8 & 221.8 & 220.7 \\ \hline \multirow{3}{*}{GS-LLFD-1} & 100 & 0.04 & 20.2 & 21.3 & 22.6 \\ \cline{2-6}  & 200 & 0.08 & 145.0 & 177.7 & 176.3 \\ \cline{2-6}  & 300 & 0.18 & 252.2 & 274.2 & 273.8 \\ \hline \multirow{3}{*}{GS-LLFD-2} & 100 & 0.36 & 33.2 & 34.8 & 34.9 \\ \cline{2-6}  & 200 & 0.17 & 111.7 & 114.9 & 118.0 \\ \cline{1-1} \cline{2-6}  & 300 & 0.39 & 238.7 & 269.7 & 263.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test results for NBSD specimens for FP-LLFD and GS-LLFD. Figure 11: Relationships between torque and yield force: (**a**) FP-LLFD; (**b**) GS-LLFD. using LLFDs, were conducted using regression Equations (3) and (4) because the LLFD theoretical design formula presented by Equation (1) had a large error. The coefficients of determination (R2) of Equations (3) and (4) had values of 0.99 and 0.98 for the FP-LLFD and GS-LLFD specimens, respectively, indicating very high correlations. \[\mathrm{V}_{d,y}=0.0013\cdot T_{0}^{2}+0.124T_{0}\ (\mathrm{forFP-LLFD}) \tag{3}\]  
\[\mathrm{V}_{d,y}=0.0022\cdot T_{0}^{2}+0.184T_{0}\ (\mathrm{forGS-LLFD}) \tag{4}\]