# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 6 Comparison of Analytical and Numerical Solutions
### Case I  
In this first case study, a fixed value is assigned to \(\rho\), \(\beta_{y}\), \(\alpha_{Q}\), and \(\alpha_{M}\), while parameter \(\alpha_{F}\) is left free in order to analyze its effect on the rotation \(\theta_{f}\) of the free end of the beam. The flexure hinge is considered in the most stressed condition, namely when \(\alpha_{Q}=0\) and \(\alpha_{M}=1\), whereas the values of \(\rho\) and \(\beta_{y}\) are chosen respectively equal to \(15\) and \(0.1\). The resulting values of \(\theta_{f}\) are collected in Table 1, where the superscript \(+\) indicates the choice of the positive sign in Equation (42). Entirely similar results but mirrored in the signs, in agreement with the absence of the transversal force \(Q\), are obtained for negative rotations, in the following indicated by the superscript \(-\). From Equation (47), the constant value of \(\theta_{f}\) resulting from the analytical solution was predictable, but the numerical solution presented in Table 1 reveals that \(\theta_{F}\) depends on \(\alpha_{F}\). In Table 1 the subscripts used in defining \(\theta_{f}\) have the following meaning: \(\theta_{f,A}\) refers to the analytical value obtained with Equation (47), \(\theta_{f,n}\), \(\theta_{f,n-\epsilon}\), \(\theta_{f,n-T}\) and \(\theta_{f,n-a(\widetilde{x})}\) are respectively the numerical solutions of the second equation in Equation (3) for the complete model, a simplified model where only the axial deformation \(\varepsilon\) is neglected, yet a simplified model where the trigonometric functions are linearized and eventually another simplified model where the effect of the axial load \(F\) is neglected. As a reference, \(\theta_{f,FEM}\) is the numerical value obtained by means of FE simulations. In addition, symbols \(\Delta_{(. )}\%\) and \(\delta_{(. )}\%\) in the tables are percentage errors defined as:  
\[\Delta_{n,A}\%=100\left|\frac{\theta_{f,n}\ -\ \theta_{f,A}}{\theta_{f,n}}\right| \tag{56}\] \[\Delta_{FEM,A}\%=100\left|\frac{\theta_{f,FEM}-\theta_{f,A}}{ \theta_{f,FEM}}\right|\] \[\delta_{\varepsilon}\%=100\left|\frac{\theta_{f,n}\ -\ \theta_{f,n- \epsilon}}{\theta_{f,n}}\right|\] \[\delta_{T}\%=100\left|\frac{\theta_{f,n}\ -\ \theta_{f,n-T}}{ \theta_{f,n}}\right|\] \[\delta_{a(\widetilde{x})}\%=100\left|\frac{\theta_{f,n}\ -\ \theta_{f,n-a( \widetilde{x})}}{\theta_{f,n}}\right|.\]  
From Table 1 and Figures 12 and 13, it can be observed that the approximation of neglecting \(\varepsilon\) and the linearization of the trigonometric functions are practically verified in  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{\alpha_{F}}\) & \(\theta_{f,A}^{+}[^{\circ}]\) & \(\theta_{f,n}^{+}[^{\circ}]\) & \(\theta_{f,n-\epsilon}^{+}[^{\circ}]\) & \(\theta_{f,n-T}^{+}[^{\circ}]\) & \(\theta_{f,n-a(\widetilde{x})}^{+}[^{\circ}]\) & \(\theta_{f,FEM}^{+}[^{\circ}]\) \\ \hline
1.00 & 12.605 & 8.740 & 8.740 & 8.731 & 12.605 & 8.744 \\
0.50 & 12.605 & 10.316 & 10.316 & 10.308 & 12.605 & 10.322 \\
0.10 & 12.605 & 12.069 & 12.069 & 12.065 & 12.605 & 12.075 \\
0.05 & 12.605 & 12.331 & 12.331 & 12.329 & 12.605 & 12.337 \\
0.01 & 12.605 & 12.549 & 12.549 & 12.549 & 12.605 & 12.555 \\ \(-\)0.01 & 12.605 & 12.661 & 12.661 & 12.662 & 12.605 & 12.668 \\ \(-\)0.05 & 12.605 & 12.892 & 12.892 & 12.894 & 12.605 & 12.898 \\ \(-\)0.10 & 12.605 & 13.192 & 13.192 & 13.197 & 12.605 & 13.199 \\ \(-\)0.50 & 12.605 & 16.209 & 16.209 & 16.264 & 12.605 & 16.218 \\ \(-\)1.00 & 12.605 & 22.582 & 22.582 & 23.010 & 12.605 & 22.593 \\ \hline \(\mathbf{\alpha_{F}}\) & \(\mathbf{\alpha_{n,A}}\%\) & \(\delta_{\varepsilon}\%\) & \(\delta_{T}\%\) & \(\delta_{a(\widetilde{x})}\%\) & \(\Delta_{FEM,A}\%\) & \\ \hline
1.00 & 44.224 & \(1.76\times 10^{-7}\) & 0.103 & 44.223 & 44.16 & \\
0.50 & 22.183 & \(1.27\times 10^{-7}\) & 0.086 & 22.183 & 22.12 & \\
0.10 & 4.446 & \(3.42\times 10^{-8}\) & 0.028 & 4.445 & 4.39 & \\
0.05 & 2.223 & \(1.78\times 10^{-8}\) & 0.015 & 2.223 & 2.17 & \\
0.01 & 0.445 & \the whole range of \(\alpha_{F}\).