# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 4 Bending Stiffness Model of a Flexure Hinge
### Solution of the Model for an Elliptical Flexure Hinge
In particular, it is:  
\[\mu_{Q,\max}=\frac{2}{3}\frac{\beta_{y}^{3/2}}{\rho^{2}}\frac{\sigma_{sn}}{E} \quad;\quad\mu_{Q,\min}=-\frac{2}{3}\frac{\beta_{y}^{3/2}}{\rho^{2}}\frac{ \sigma_{sn}}{E}\ . \tag{43}\]  
A look at Equation (42) reveals that its expression tends to Equation (40) when the second term under the square root is negligible with respect to the first; namely, when:  
\[|\mu_{Q}|<<\frac{2}{3}\frac{\beta_{y}^{3/2}}{\rho^{2}}\frac{\sigma_{sn}}{E}. \tag{44}\]  
Therefore, the flexure is the more stressed at its geometric center; the smaller \(\mu_{Q}\) is in modulus. A final investigation can be directed towards the search for the expressions and trends of \(K\) and \(\Delta M\), presented above. By calculating the integrals in Equation (16) and expressing them in terms of the nondimensional parameters \(\rho\), \(\beta_{y}\) and \(Q\), it is:  
\[K=\frac{4\ E\ \alpha_{x}^{2}\ l}{3\ \rho^{3}}\frac{\delta_{y}\ (\beta_{y}\ \gamma_{y})^{\frac{5}{2}}}{6 \sqrt{\beta_{y}\gamma_{y}}+8\sqrt{\beta_{y}^{3}\gamma_{y}}+4\sqrt{\beta_{y}^{ 3}\gamma_{y}}+3\delta_{y}^{2}\big{[}\pi+2\csc^{-1}(\delta_{y})\big{]}}\ . \tag{45}\]  
\[\sigma(x)=-E\;\kappa(x)\;y(x), \tag{38}\]0  
It can be observed from Equation (45) that, except for the Young's Modulus, coefficient \(K\) depends only on the beam geometry, whereas the dependence on the loads can be found only in \(\Delta M\). The stiffness coefficient \(K\) mainly depends on the beam geometry. Its trend is shown in Figure 11 when \(\beta_{y}\) and \(\rho\) vary. In addition, it can be observed that in order to have a low value of stiffness \(K\), which corresponds to a large rotation without excessive tension of the material, high values of \(\rho\) are preferred and, conversely, low values of \(\beta_{y}\). Regarding parameter \(\Delta M\), it is easy to verify that when it is null, the classical linear relation between \(M_{f}\) and \(\theta_{f}\) is obtained, namely: \(M_{f}=K\theta_{f}\). Moreover, by looking at the second expression in Equation (45), the right side of the equation represents the first addendum in Equation (42). With reference to the results presented in Equations (42) and (44)  
\[\sigma(x)=-E\;\kappa(x)\;y(x), \tag{38}\]1 Trend of function \(K\) in terms of \(\beta_{y}\) and \(\rho\). and remembering the definition of \(\mu_{Q}\), the following expressions of \(M_{f}\) and \(Q\) can be imposed introducing the two coefficients \(\alpha_{Q}\) and \(\alpha_{M}\):  
\[ \begin{split} Q=\alpha_{Q}Ela_{x}\mu_{Q,\max}\\ M_{f}=\alpha_{M}M_{f,\max}\end{split}  \tag{46}\]  
where \(\alpha_{M}\in[0;1]\) and \(\alpha_{Q}\in[-1;1]\). The smaller \(\alpha_{Q}\) is in modulus, the more the maximum stress is in the middle section of the flexure joint. On the other hand, the smaller \(\alpha_{M}\) is, the further away from the yield condition. An analytical study of \(\theta_{f}\) allows us to evaluate its maximum value. In particular, it occurs when \(\alpha_{M}=1\) and \(\alpha_{Q}=0\), namely for a pure bending load:  
\[\theta_{f,\max}=\pm\frac{2}{3}a_{x}^{2}l\left(\frac{\beta_{y}}{\rho}\right)^ {2}\frac{\sigma_{sn}}{K}. \tag{47}\]