## 2 In-Plane Bending of Cantilever Beam
The exact value comes from Equation (15) and equals \(\sigma_{x}(L/2,-c)=400\) MPa. It also provides the reference to estimate the relative error \(\varepsilon_{\sigma{\rm X}}\). The stress values, together with their counterparts of the error, show that the meshes made of triangular FEs, i.e., M-T3, R-T3 sh, and R-T3m, provide deficient responses. The exception is the finest mesh. Only the results for case M-T3 dr, when drilling rotations are activated in Midas, can be treated as satisfactory. On the other hand, if Q4 FEs are applied, the values of stress \(\sigma_{x{\rm C}}\) are much closer to the analytical solution. However, the error in the results in the case of M-Q4 and R-Q4m for the coarsest mesh exceeds 25%. In the next figure, Figure 5, the attention is focused on tangential stress \(\tau_{xy}\) at point D, called here shortly \(\tau_{\rm D}\). The absolute value \(|\tau_{xy}(L/2,0)|=50\) MPa connected with Equation (16) is taken as the analytical solution to assess the relative error \(\varepsilon_{\tau\,\mathrm{D}}\). Now the poorest results are noticed for cases M-Q4 and R-Q4m if the computations are performed with coarse or medium meshes. On the other hand, the values of stress \(\tau_{\mathrm{D}}\) for M-Q4 dr, R-Q4 sh, and M-T3 dr are nearest to the theoretical 50 MPa. It is also seen for these cases that convergence is attained pretty quickly. Summarizing all the results for displacements and stresses obtained so far, it can be stated that, using coarse meshes, the values are more or less underestimated. Of course, this is related to the shear locking effect for membranes, which occurs for poor discretizations and when the displacement field as the primary one is interpolated in FEs in the most conventional way. Figure 6 depicts the values of normal stress \(\sigma_{x}(L,-c)\) at point E, which is called \(\sigma_{x\mathrm{E}}\), and their associated relative errors \(\varepsilon_{x\mathrm{E}}\). The tendency for successive meshes seems to be similar to \(\sigma_{x\mathrm{C}}\). Stress values \(\sigma_{x\mathrm{E}}\) for M-T3, R-T3 sh, and R-T3m in the case of a coarse mesh are strongly underestimated, while for M-T3 dr and for the solutions with FEs Q4, this underestimation is less pronounced and decreases quickly. However, the values of \(\sigma_{x\mathrm{E}}\) for the two finest meshes exceed the theoretical value of 800 MPa determined in Equation (6). Moreover, it is visible that relative errors \(\varepsilon_{x\mathrm{E}}\) are larger for the mesh with FE size 0.75 cm than for the mesh with FE size 1.5 cm. It should be noted that point E is located at the corner of the specimen on the fixed edge. A tendency towards infinite stress with mesh refinement is detected in such regions during numerical analyses, which is also in accordance with the classical elasticity theory. In other words, the value of stress \(\sigma_{x\mathrm{E}}\) can grow and grow when discretization becomes increasingly dense. This phenomenon is known as stress singularity; see [18; 19; 32]. From a physical point of view, stress concentration can occur in areas with geometric or material discontinuity, and it destroys the results of simulations with fine discretization. Moreover, the stress field is extrapolated from integration points that are located inside FEs, and this can additionally amplify the increase in stress values at such corners. Due to this fact, the convergence study or the search for maximum stress in the singularity region cannot lead to an objective estimation. Here, point E is selected deliberately to show that performing a convergence study does not make sense for certain locations. Furthermore, a so-called root singularity pattern, shown as a contour plot in [17], confirms that the highest intensity of the stress field arises exactly at corners near the clamped edge. The issue of stress singularities is extensively discussed in [33]. In FEM, special singular elements and a measure called the stress intensity factor can be introduced; see [34].