## 4 Discussion  
The aims of this study are to validate the effectiveness of FE simulation of the FLBS of a shoe sole and to analyze the effect of different loading rates on the stiffness by combining both experimental and simulation methods. Additionally, this study investigates how varying carbon plate thicknesses and forefoot curvature radii affect the FLBS of both the carbon plates and the 'plate-sole' under identical loading conditions, ultimately clarifying the relationship between the carbon plate and the 'plate-sole' in terms of stiffness. In this study, we established different rate variables in combination with experimental and simulation analysis and found that varying the rate within the same displacement range has no significant effect on the stiffness results. This implies that in the simulation model, the rate can be freely set within the same displacement range for stiffness testing. In previous studies, the three-point bending test has often been used to evaluate the FLBS of shoe soles, especially for carbon plate running shoes, and to qualitatively investigate both the impact of carbon plates with different stiffness levels on running economy and the effect of stiffness values on lower extremity biomechanics [9]. However, there is no unified testing standard for the current shoe sole three-point bending test. The initial loading method involved setting the stamping head to move 7.5 mm vertically downward at a rate of 75 mm/s over 0.1 s, with the center of the forefoot as the reference point [9]. Subsequently, the loading rate was adjusted in various studies to 15 mm/s [7], 500 mm/min (8.33 mm/s) [20], 16 mm/s [36], 4 mm/s [19], 12.5 mm/s [27], and 10 mm/s [26], with displacement ranges including 10 mm [18], 20 mm [19], and 15 mm [6]. Although the measurement ranges differ, all studies performed linear regression on the linear portion of the force-displacement curve [6; 26; 37]. By employing FE analysis to simulate three-point bending tests, the present study provides a theoretical foundation for further exploring the mechanical properties of carbon plate configurations in footwear. The results of this study indicate that increased carbon plate thickness leads to a significant rise in the stiffness of both the carbon plate and the 'plate-sole'. However, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Plate** & **Curvature** & **Reduction** & **Curvature** & **Plate** & **Reduction** \\
**Thickness (mm)** & **Radius (mm)** & **Percentage (\%)** & **Radius (mm)** & **Thickness (mm)** & **Percentage (\%)** \\ \hline \multirow{3}{*}{\(1\)} & 200 & 27.86 & & 1 & 27.86 \\  & 250 & 23.85 & 200 & 1.5 & 56.61 \\  & 300 & 17.29 & & 2 & 72.69 \\ \hline \multirow{3}{*}{\(1.5\)} & 200 & 56.61 & & 1 & 23.85 \\  & 250 & 54.31 & 250 & 1.5 & 54.31 \\  & 300 & 50.41 & & 2 & 70.75 \\ \hline \multirow{3}{*}{\(2\)} & 200 & 72.69 & & 1 & 17.29 \\  & 250 & 70.75 & 300 & 1.5 & 50.41 \\ \cline{1-1}  & 300 & 68.37 & & 2 & 68.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of stiffness reduction percentages of carbon plate vs. ‘plate-sole’ for same thickness and curvature radius. when comparing the two, it can be observed that the increase in 'plate-sole' stiffness is less pronounced than that of the carbon plate. In previous studies, increasing the number of stacked carbon plate layers is a common method to enhance the longitudinal bending stiffness of experimental shoe soles [17]. [PERSON] [38] embedded varying numbers of 1 mm thick flat carbon fiber plates into misdoles, while [PERSON] [39] embedded 1 mm and 1.5 mm thick carbon fiber plates in a shoe's midsole. In experimental testing, changing stiffness by stacking layers presents challenges such as imperfect bonding between the carbon plates and the complex manufacturing process of embedding them into the shoe sole. However, using models for analysis allows for easy adjustment of the carbon plate thickness and the positioning of the embedded plates in the sole, addressing these limitations.