# Insights into Bending Stiffness Modeling of Elementary Flexure Hinges
## 2 Nonlinear Bending of a Straight Beam  
### Large Displacement Model of a Planar Beam  
This work starts from a straight axis Euler-Bernoulli beam of length \(L\) in the plane \(xy\), as shown in Figure 2 on the left, aligned with the \(x\)-axis. It follows that all results shown  
Figure 1: Chart of the most frequent geometries of flexures. below refer to beams that meet the assumptions of Euler-Bernoulli's model. Furthermore, the study of flexures characterized by homogeneous and isotropic material under nominal temperature conditions is addressed. The undeformed state of the beam represents the reference condition, with points \(A\) and \(B\) being its two ends. Under static loads acting on the mentioned plane the beam changes geometry and assumes a different shape, where the new positions \(A^{\prime}\) and \(B^{\prime}\) of its extremes result in accordance with the planar load condition shown in Figure 3, namely a force \(R_{0}\) in a generic direction of the plane \(xy\) and a bending moment \(M_{f}\) about its orthogonal \(z\)-axis applied at point \(B\), or more generally at point \(B^{\prime}\) after the beam deformation. A generic infinitesimal element \(dx\) in the beam reference condition deforms into the element \(ds\), which assumes the new coordinate position \(\zeta\) and \(\psi\). The two differentials \(dx\) and \(ds\) are related by the following relationship: \(ds=dx(1+\varepsilon)\), where \(\varepsilon\) is the axial deformation of the beam. Be \(M\) the axial bending moment, \(E\) the Young's modulus, \(I_{z}\) the moment of inertia of the beam section, and \(\kappa\) the beam curvature. Two definitions of curvature can be found in the literature: the mechanical curvature and the geometric curvature, denoted as \(\kappa_{m}\) and \(\kappa_{g}\) respectively [43]:  
\[\kappa_{m}=\frac{d\theta}{dx} \tag{1}\] \[\kappa_{g}=\frac{d\theta}{ds}=\frac{d\theta}{dx}\frac{dx}{ds}= \frac{\kappa_{m}}{1+\varepsilon}\]  
In the scientific literature, \(\kappa_{m}\) is commonly used [44; 45; 46] for many reasons, in particular for its simplicity and its feature of isolating the effect of pure bending from curvature  
Figure 3: Static load conditions. Figure 2: Undeformed and deformed states of a straight axis beam (**left**) and load conditions (**right**). variations produced by stretching. However, in other works [47; 48; 49]\(\kappa_{g}\) is preferred over \(\kappa_{m}\) because it is considered the most correct curvature due to its geometric meaning. With reference to Figure 3, a static analysis of a portion of the beam, from the generic section \(P^{\prime}\) to its end \(B^{\prime}\) where external loads are applied, provides the moment equation with respect to \(ds\) or \(dx\):  
\[ \begin{split}\frac{dM}{ds}&=R_{0}\sin\left(\theta- \varphi\right)\\ \frac{dM}{dx}&=(1+\varepsilon)R_{0}\sin\left(\theta -\varphi\right)\end{split} , \tag{2}\]  
where \(\varphi\) is the slope of vector \(R\) of magnitude \(R_{0}\) with respect to the \(x\)-axis. Exploiting the curvature-moment relationship and the definition of curvature, the following differential equations can be found:  
\[\begin{split}\theta^{\prime\prime}(x)+\frac{I_{z}^{\prime}(x)}{ I_{z}(x)}\theta^{\prime}(x)=\frac{R_{0}}{EI_{z}(x)}[1+\varepsilon(x)]\sin\left[ \theta(x)-\varphi\right]&(\kappa=\kappa_{  
On the basis of all these assumptions, the following simplified form is obtained:  
\[\widetilde{\theta}^{\prime\prime}(\widetilde{x})+\frac{\widetilde{I}_{z}^{ \prime}(\widetilde{x})}{\widetilde{I}_{z}(\widetilde{x})}\;\widetilde{\theta}^ {\prime}(\widetilde{x})+a(\widetilde{x})\widetilde{\theta}(\widetilde{x})=b( \widetilde{x}), \tag{8}\]  
where  
\[ \begin{split} a(\widetilde{x})&=-\frac{Ax_{0}^{2}}{EI_ {20}\widetilde{I}_{z}(\widetilde{x})}\cos(\varphi)\\ b(\widetilde{x})&=-\frac{Ax_{0}^{2}}{EI_{20} \widetilde{I}_{z}(\widetilde{x})}\sin(\varphi)\end{split} . \tag{9}\]  
An equation with variable coefficients, such as Equation (8), admits an analytical solution when the zero-order derivative term is identically null.