## 2 In-Plane Bending of Cantilever Beam
Figure 5: Values of (**a**) tangential stress and (**b**) relative error at point D for all cases computed in Midas and Robot (detailed explanation of abbreviations in Table 1). Figure 6: Values of (**a**) normal stress and (**b**) relative error at point E for all cases computed in Midas and Robot (detailed explanation of abbreviations in Table 1). M-T3 (blue line with filled triangles) and R-T3m (purple line with blank inverted triangles) are poor for coarse and medium meshes. There are responses for discretizations with FEs known as constant strain triangles (CST), where standard bilinear shape functions are implemented. On the other hand, \(u_{\rm B}\) for cases M-Q4 dr and R-Q4 sh is the best, so the smallest error \(\varepsilon_{u{\rm B}}\) is visible for them, even if the coarse mesh is used in the computations. Moreover, both of them have two identical numerical values for each corresponding mesh, so they are marked as one red line with filled squares. A similar coincidence as for the aforenamed cases with quadrilaterals Q4 can also be seen for cases with triangle elements T3. Despite the fact that the displacements for corresponding meshes are not exactly the same, the diagrams for cases M-T3 dr (green line with filled inverted triangles) and R-T3 sh (red line with blank triangles) overlap almost perfectly. It can be concluded that when in-plane bending is analyzed, membrane FEs in Midas with active drilling rotations behave very similarly to shell FEs in Robot. As expected, it is shown that the values of displacement \(u_{\rm B}\) and the relative error determined for them in all cases with triangle meshes reveal lower accuracy. However, for both types of FEs, quadratic convergence is attained as the mesh is refined. Analogically to M-Q4 dr and R-Q4 sh, the results for cases M-Q4 and R-Q4m are numerically twinned, so again, only one diagram is plotted (the purple line with blank squares). These responses for displacement \(u_{\rm B}\) on the left and the relative error on the right are given for solutions with classic four-noded FEs (Q4) without any enhancements. For coarser meshes, especially with triangular FEs, it is observed that the presence of drilling rotations in the finite element interpolation improves the solution. Figure 3 illustrates the results observed at point 0. Values of vertical translation, also interpreted as the beam deflection and marked by \(v_{0}\), are represented on the left. Numerical solutions are confronted with the theory, similar to the analysis discussed in Figure 2. Here, the thin black solid line corresponds to the value \(v^{\rm B-E}(0,0)\) given in Equation (5) for the Bernoulli-Euler theory, while the thick black solid line conforms to the value \(v^{\rm T-E}(0,0)\) determined in Equation (9) according to the Timoshenko-Ehrenfest theory. Similarly to Figure 2, a gray zone between the lines is apparent in Figure 3, creating an interval based on both theories as a reference for values \(v_{0}\). On the right-hand side in Figure 3, the relative error \(\varepsilon_{v0}\) is reproduced, but here in regard to \(v^{\rm T-E}(0,0)=0.542\) cm. Generally, it can be stated that all comments that describe the results given in Figure 2 are also valid for the current results. Similarities and analogies between respective cases are the same. One can only add that displacement \(v_{0}\) for cases M-Q4 dr and R-Q4 sh are within the zone of analytical solutions when the square mesh with FE size 6 cm or denser is applied. On the contrary, the numerical value of \(v_{0}\) for M-T3 and R-T3m with triangular FEs is within this zone only when the size 0.75 cm is introduced. Based on the results presented in Figures 2 and 3, it can also be concluded that displacements converge from below in this benchmark. The results for stresses are shown in Figures 4-6. The method of presentation and description for subsequent cases is the same as previously. Figure 4 refers to the normal stress \(\sigma_{x}\) observed at point C, so it is called here \(\sigma_{x{\rm C}}\).