# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Effective Stress Maps  
In this section, the effective stress distribution maps for both plate configurations, simply supported and clamped plate, and both numerical method, FEM and RPIM, will be presented. The FEM and RPIM results were once again obtained assuming the full Newton-Raphson nonlinear solution algorithm (KT-All). After extrapolating the stress state from the integration points near the plate's top layer to its closest nodes (belonging to the plate's top surface), the (extrapolated) von Mises equivalent stress on each of those nodes was calculated:  
\[\bar{\sigma}_{eq}=\sqrt{\frac{1}{2}\Big{[}(\sigma_{xx}-\sigma_{yy})^{2}+(\sigma _{yy}-\sigma_{zz})^{2}+(\sigma_{zz}-\sigma_{xx})^{2}+6(\sigma_{yz}^{2}+\sigma_ {xx}^{2}+\sigma_{xy}^{2})\Big{]}}. \tag{62}\]  
The obtained FEM von Mises equivalent stress maps for the simply supported plate are presented in Figure 10, and the RPIM results are shown in Figure 11. The load increment is in [kN/m\({}^{2}\)], and the maximum and minimum stress values of each incremental load (subfigure) are the ones indicated in the caption of the corresponding subfigure. It is possible to observe that both FEM and RPIM produce very similar results. As expected, the simply supported plate forms a rupture line along the plate diagonal, connecting the plate's centre and the plate's corner. Notice that the material enters the plastic regime along that diagonal line first, and then expands from that line to the remaining area as the load increment increases. Regarding the clamped plate results, the corresponding von Mises equivalent stress maps are presented in Figures 12 and 13 for FEM and RPIM, respectively. Both FEM and RPIM solutions are again very similar. It is clear that the initiation of material yields at the boundaries of the plate, and then it starts to yield at the centre of the plate. Afterward, the yielding advances to a diagonal line merging the boundary yield material to the yield material of the plate's centre. Both the rupture paths observed in the simply supported and clamped plates are the ones described in the literature for these two benchmarks [26]. Figure 9: Stress distribution along the thickness in line D-D\({}^{\prime}\) for the clamped. FEM analysis (**a**) \(\sigma_{xx}\) and (**c**) \(\tau_{xz}\). RPIM analysis (**b**) \(\sigma_{xx}\) and (**d**) \(\tau_{xz}\). Load increments in [kN/m\({}^{2}\)]. Figure 10: Effective von Mises stress distribution maps obtained with FEM for five load increments considering the simply supported plate. For each load increment, the minimum and maximum effective stresses are: (**a**) \(f=814.89\)\(\mathrm{K}\mathrm{N}\mathrm{/}\mathrm{m}^{2}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=170\) MPa. (**b**) \(f=998.25\)\(\mathrm{K}\mathrm{N}\mathrm{/}\mathrm{m}^{2}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=205\) MPa. (**c**) \(f=1181.60\)\(\mathrm{K}\mathrm{N}\mathrm{/}\mathrm{m}^{2}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=210\) MPa. (**d**) \(f=1364.95\)\(\mathrm{K}\mathrm{N}\mathrm{/}\mathrm{m}^{2}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=215\) MPa. (**e**) \(f=1487.18\)\(\mathrm{K}\mathrm{N}\mathrm{/}\mathrm{m}^{2}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=220\) MPa. Figure 11: Effective von [PERSON] stress distribution maps obtained with RPIM for five load increments considering the simply supported plate. For each load increment, the minimum and maximum effective stresses are: (**a**) \(f=807.13\)\(\mathrm{KN/m^{2}}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=170\) MPa. (**b**) \(f=988.79\)\(\mathrm{KN/m^{2}}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=205\) MPa. (**c**) \(f=1170.34\)\(\mathrm{KN/m^{2}}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=210\) MPa. (**d**) \(f=1351.95\)\(\mathrm{KN/m^{2}}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=215\) MPa. (**e**) \(f=1473.02\)\(\mathrm{KN/m^{2}}\): \(\sigma_{min}=0\) MPa; \(\sigma_{max}=220\) MPa. Figure 12: Effective von Mises stress distribution maps obtained with FEM for five load increments considering the clamped plate.