# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## Numerical Results
### Trial Functions  
With reference to Section 2.3, the in-plane trial functions \((U_{j}(s),V_{j}(s))\) are analytically evaluated as the in-plane normal modes of a semi-annular inextensible Euler-Bernoulli beam matching the shape of the cantilever cross-section (Figure 3a), without any external constraints. Differently, in case of cross-sections with general expression of the curvature as a function of \(s\), numerical techniques might be called for. Exclusively for this step, the mass per unit length \(m\) and the bending stiffness \(EI\) are  
Figure 3: Cross-section and load conditions in case-studies: (**a**) load condition 1; (**b**) load condition 2.  
assumed uniform and, conventionally, their values taken as \(m=1\) kg/m and \(EI=1\) Nm\({}^{2}\). The normal modes of the arch beam turn out to be the solutions of the following eigenvalue problem (see [34]):  
\[\frac{dU}{ds}-\bar{k}V=0 \tag{33}\] \[\frac{dV}{ds}+\bar{k}U-\Theta_{b}=0\] (34) \[k_{b}=\frac{d\Theta_{b}}{ds}\] (35) \[M=E\bar{I}k_{b}\] (36) \[\frac{dN}{ds}-\bar{k}T+m\lambda U=0\] (37) \[\frac{dT}{ds}+\bar{k}N+m\lambda V=0\] (38) \[\frac{dM}{ds}+T=0 \tag{39}\]  
with natural boundary conditions in \(s=0\), \(\pi R\):  
\[N =0\] \[T =0 \tag{40}\] \[\frac{dU}{ds}-\bar{k}V=0 \tag{33}\]0  
where the above represent the normal and shear forces and bending moment, respectively. In particular, Equation (33) states the vanishing of the axial strain of the arch beam, which corresponds to [PERSON]'s condition of profile mid-plane inextensionality (see Equation (23) for comparison); Equation (34) states the vanishing of arch beam shear strain, where \(\Theta_{b}\) is the rotation of the cross-section of the arch about \(\mathbf{e}_{b}\)--this condition corresponds to former [PERSON]'s assumption regarding wall bending. Equation (35) defines the bending curvature \(k_{b}\) about \(\mathbf{e}_{b}\), Equation (36) is the classical constitutive law for bending moment \(M\) and curvature \(k_{b}\), and Equations (37)-(39) are the balance equations for normal \(N\) and shear \(T\) forces, and bending moment \(M\), respectively. The eigenvalue \(\lambda\) is the square of the circular frequency \(\omega\). It is worth noting that no constitutive law is used for \(N\) and \(T\), they being the internal reactive forces to the constraints (33) and (34). Assuming an exponential solution of type \(\mathbf{v}(s)=\hat{\mathbf{v}}e^{\Omega s}\) where \(\mathbf{v}(s)=(U,V,\Theta_{b},k_{b},N,T,M)^{T}\), after usual steps in modal analysis, the following bi-cubic dispersion equation is obtained:  
\[\frac{dU}{ds}-\bar{k}V=0 \tag{33}\]1  
Its roots \(\Omega_{i}(\lambda)\), \(i=1,\dots,6\), after substitution in Equations (33)-(39), allow one to evaluate the relevant eigenvectors \(\hat{\mathbf{v}}_{i}\) and build the solution \(\mathbf{v}(s)=\sum_{i}c_{i}\hat{\mathbf{v}}_{i}e^{\Omega s}\), where \(c_{i}\) are six undetermined constants. The latter expression is substituted in Equation (40) to get the first \(K\) natural frequencies (or \(\lambda\)) and, for each of them, five of the six coefficients \(c_{i}\), while the sixth is given by the normalization condition. As expected, \(\lambda=0\) is a solution of multiplicity 3, giving rise to three in-plane independent rigid motions. An additional rigid motion, required to describe the uniform displacement in direction \(\mathbf{e}_{x}\), is prepended to them as reported at the end of Section 2.3. Once the modal in-plane components \(U_{j}(s)\), \(V_{j}(s)\), \(j=1,\dots,K\) are obtained, the out-of-plane ones \(W_{j}(s)\) are evaluated as well, as described in Section 2.3. The first four modes, relevant to rigid motions, are shown in Figure 4 where, in the first and second row, the in-plane and out-of-plane components are drawn, respectively.