# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Stress Diagrams through the Plate's Thickness  
In this section, the stress distributions along with the plates' thickness for distinct increment levels, and for the three numerical methods, will be presented. These results were  
Figure 4: Normalized load-transversal displacement curves obtained with three nonlinear solution algorithms (KT0, KT1, and full Newton–Raphson method—KT-ALL). FEM analysis for the (**a**) simply supported plate and (**b**) the clamped plate. RPIM analysis for the (**c**) simply supported plate and (**d**) the clamped plate. obtained only with the KT-All nonsolution algorithm. As for units, the load increments are in [kN/m2], the stress value (horizontal axis) in MPa, and the thickness coordinate (vertical axis) is in meters. First, the stress distribution along line A-A\({}^{\prime}\) is shown for the simply supported plate. Thus, in Figure 5, the FEM and RPIM results are shown side by side. As a first observation, it is possible to visualise that the stress distribution along the thickness for the normal stresses \(\sigma_{xx}\) and \(\sigma_{yy}\) are virtually identical. Such an observation indicates that the symmetry was properly implemented. Additionally, the normal stresses  
Figure 5: Stress distribution along the thickness in line A-A\({}^{\prime}\) for the simply supported plate. FEM analysis (**a**) \(\sigma_{xx}\), (**c**) \(\sigma_{yy}\), (**e**) \(\tau_{xz}\) and (**g**) \(\tau_{yz}\). RPIM analysis (**b**) \(\sigma_{xx}\), (**d**) \(\sigma_{yy}\), (**f**) \(\tau_{xz}\) and (**h**) \(\tau_{yz}\). Load increments in [kN/m2]. obtained with RPIM are very similar to the ones obtained with the FEM. It is possible to visualise that, as the material starts to yield, the stress distribution along the thickness leaves the linear behaviour and starts showing a pronounced curve at the top and bottom layers, consistent with the maximum (yield) stress allowed. The most significant differences can be found in the shear components of the stress tensor: \(\tau_{xz}\) and \(\tau_{yz}\). However, the differences become smaller as the load increment grows. At the centre of the plate, along the thickness, the distribution of shear stresses \(\tau_{zz}\) and \(\tau_{yz}\) should be zero. This is true for the first increment (around 807-815 kN/m\({}^{2}\)). Figure 5e-h does not show such a null line for the first increment because the stress on the nodes is extrapolated from the integration points in its vicinity (which do not belong exactly to the centre of the plate, and therefore possess non-null shear stresses). Afterward, yielding starts to occur at the centre of the plate and the stress field of the integration points along the thickness starts to increase due to the reconfiguration of the stress, a consequence of the stress-returning algorithm. For the clamped plate, four interest lines along the plate thickness are analysed. Thus, Figures 6-9 represent the stress distributions obtained along lines A-A\({}^{\prime}\), B-B\({}^{\prime}\), C-C\({}^{\prime}\) and D-D\({}^{\prime}\), as indicated in Figure 3, respectively. Observing Figures 6-9 it is noticeable that the load increments of FEM and RPIM are different. Such differences come from the elastoplastic algorithm implemented by the authors. The algorithm first identifies the load increment corresponding to the problem's elastic limit, \(f_{0}=f_{el}\); then such a load increment is multiplied by a factor, which in this work was 2.5, and an ultimate load is achieved: \(f_{u}=2.5\times f_{el}+f_{el}\). Then, the number of increments is decided. In this work, 10 increments were used for each elastoplastic analysis. Thus, after the elastic limit, each incremental load will have the following magnitude: \(f_{i}=2.5\times f_{el}/10\). Hence, if two distinct numerical methods predict two distinct elastic limits (because distinct numerical methods produce distinct numerical results), the ultimate load, and consequently the incremental load magnitude, of each numerical method will also be different.