# Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections
## Numerical Results
### Equilibrium Configuration  
Using the \(K=8\) trial functions discussed in Section 3.2, the equilibrium Equation (29) can be evaluated through Equation (27), and combined to the boundary conditions (30) relevant to the cantilever, i.e., geometric at \(x=0\) and natural at \(x=l\). The obtained equations are then solved using a finite difference code based on the three-stage Lobatto IIIa formula, as implemented in the bvp4c library command in MATLAB [35], and the solution in terms of \(a_{j}(x)\), \(j=1,\ldots K\), is used to calculate displacement and stress components in specific points of the thin-walled beam. In particular, for the two load conditions, plots are shown for: kinematic descriptors \(a_{j}(x)\) as functions of \(x\in[0,l]\); displacement components \(u_{x},u_{y},u_{z}\) as functions of \(x\) for three points of the mid-curve \(\bar{\gamma}\), i.e., at positions where \(s=0\) (referred to as position S, i.e., the most southerly point), \(s=\frac{\pi R}{2}\) (referred to as position W, i.e., the most westerly point), \(s=\pi R\) (referred to as position N, i.e., at the most northern point)--see also Figure A1 in Appendix A for reference; stress component \(\sigma_{x}^{M}\) as a function of \(x\) at the same three positions (S,W,N); stress component \(\sigma_{x}\) as a function of \(n\in[-\frac{b}{2},\frac{b}{2}]\)  
Figure 4: In-plane (ip) and out-of-plane (op) components of the trial functions, from I to IV (rigid modes). Blue dashed line: initial configuration; black solid line: trial functions. Figure 5: In-plane (ip) and out-of-plane (op) components of the trial functions, from V to VIII (deformation modes). Blue dashed line: initial configuration; black solid line: trial functions. at \(x=0\) in the same three positions. The GBT framework analytical solutions for \(u_{x},u_{y},u_{z}\) are compared to the outcomes of a FEM model implemented in the Abaqus software (dotted lines in the plots). Details on the numerical model are given in Appendix A. For the load condition 1 (load in \(\mathbf{e}_{z}\) direction), the solution is shown in Figure 6. From Figure 6a,b it can be observed that the component \(a_{1}\), corresponding to uniform extension, is always zero, as expected in a linear problem with structure subjected to transverse loads only. Moreover, amplitudes \(a_{j}\), \(j=3,\ldots,6\) have at least one order of magnitude larger than the remaining \(a_{2},a_{7},a_{8}\), revealing that global displacement along \(\mathbf{e}_{y}\) is almost zero and higher modes are actually not required for this case (it is verified, even if not reported here for the sake of brevity, that higher modes than \(K=8\) give vanishing contributions for the two analyzed cases). In particular, the components \(a_{3},a_{4},a_{5}\) are those which give larger contributions, indicating that besides a rigid motion of the cross-section, a not-negligible deformation component related to \(a_{5}\) (and in minor intensity \(a_{6}\)) is present. In Figure 6c-e, the individual displacement components at the three points of the cross-section (S,W,N) are shown: the component \(u_{x}\) comes only from warping of the cross-section and is much smaller than \(u_{z}\) and \(u_{y}\). Significant in-plane rotation (about \(\mathbf{e}_{x}\)) and change in shape of the cross-section are observed, as related to quite large \(a_{4}\) and \(a_{5}\), respectively, as well as the expected slightly larger maximum displacement in direction \(\mathbf{e}_{z}\) (direction of the load). These outcomes reveal very good agreement with the FEM results.