# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Stress Diagrams through the Plate's Thickness
Regarding line A-A\({}^{\prime}\), Figure 6, it is possible to visualise that FEM and RPIM produce similar results for similar load increment levels. Additionally, as already observed in the simply supported plate, the normal stresses \(\sigma_{xx}\) and \(\sigma_{yy}\) evolve from a linear elastic distribution along the thickness to a curved yield configuration at the top and bottom layers of the plate. The shear stresses \(\tau_{xz}\) and \(\tau_{yz}\) start with a value very close to zero (it is not zero due to the same reason already discussed for the simply supported plate) and both evolve to a bilinear configuration, with lower shear stress values in the top and bottom layers of the plate and maximum stress in the plate's middle layer. Such stress evolution is due to the material yielding and the consequent required stress returning procedure. Along line B-B\({}^{\prime}\), Figure 7 shows that the normal stresses \(\sigma_{yy}\) obtained with FEM and RPIM are close for similar incremental load levels. However, the shear stress results are very different. The FEM obtains an expected shear stress \(\tau_{yz}\) distribution, showing an unsymmetrical irregular path. In opposition, the RPIM produces the expected result, a parabolic distribution, symmetric with respect to the plate's middle layer, that becomes gradually shaped (with a central peak at the plate's middle layer) as the material yielding evolves. Line C-C\({}^{\prime}\) was selected to study the variation of the shear stress \(\tau_{xy}\). Being the plate's corner, shear stresses \(\tau_{xy}\) are at maximum at this location. Hence, Figure 8 presents, along the plate's thickness, the evolution of shear stress \(\tau_{xy}\) with the plate's yielding. At first sight, the most evident observation is the difference between FEM and RPIM results. The shear stress \(\tau_{xy}\) produced with FEM maintains its almost linear distribution along the thickness as the plate yields. On the other hand, with RPIM for the first load increments, \(\tau_{xy}\) maintains a relatively low magnitude; however, as the material starts to yield at this location, the stress level rises rapidly. Concerning the stress distribution along the line D-D\({}^{\prime}\), Figure 9, the FEM and RPIM results are consensual. The normal stress \(\sigma_{xx}\) and the shear stress \(\tau_{xz}\), obtain with FEM and RPIM, are very close. Furthermore, along this line D-D\({}^{\prime}\), the shear stress \(\tau_{xz}\) obtained with FEM follows the same evolution as the one observed with RPIM as the plate yields. Figure 6: Stress distribution along the thickness in line A-A’ for the clamped. 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 \([\mathrm{kN/m^{2}}]\). Figure 8: Stress distribution along the thickness in line C-C’ for the clamped. FEM analysis (a) \(\tau_{xy}\). RPIM analysis (b) \(\tau_{xy}\). Load increments in [kN/m2]. Figure 7: Stress distribution along the thickness in line B-B’ for the clamped. FEM analysis (a) \(\sigma_{yy}\) and (c) \(\tau_{yz}\). RPIM analysis (b) \(\sigma_{yy}\) and (d) \(\tau_{yz}\). Load increments in [kN/m2]. Figure 9: _Cont._