# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Normalized Central Displacement  
In this section, the \(w\) displacements of point \(P\) (obtained along \(Oz\)) as a function of the applied load will be presented. Thus, in order to represent normalized load-displacement curves, the obtained \(w\) displacement are normalized with  
\[\bar{w}_{P}=\frac{w_{P}\cdot D}{M_{P}\cdot L^{2}}, \tag{58}\]  
and the force is normalized with  
\[\bar{f}=\frac{f\cdot L^{2}}{M_{P}}, \tag{59}\]  
being,  
\[D=\frac{100\cdot E\cdot h^{3}}{12(1-\
u^{2})}, \tag{60}\]  
and  
\[M_{P}=\sigma_{Y}\frac{h^{2}}{4}, \tag{61}\]  
where \(w_{P}\) is the obtained transversal displacement (along \(Oz\)) on point \(P\), \(h\) is the thickness of the plate, \(E\) the elasticity (Young's) Modulus, \(\
u\) the Poisson's coefficient, \(M_{P}\) the plastic moment, and \(\sigma_{Y}\) the yield stress of the material being used.  
The normalized load-transversal displacement curves are presented in Figure 4 for both simply supported and clamped plates, and for both numerical methods (FEM and RPIM).  
Regarding the simply supported plate, it can be observed that by using the KT1 and KT-All nonlinear solution algorithms allows similar solutions for FEM and RPIM. Comparing the results with the literature, it is perceptible that both FEM and RPIM (using the KT-All nonlinear solution algorithm) are capable of achieving close results with the EFGM formulation by using an equivalent single layer theory [26].  
Figure 3: Square plate representation showing the geometric properties, the symmetry assumption and the uniform distributed load considered in all analyses.  
For the clamped plate, Figure 4b shows that the FEM with KT1 and KT-All is capable of delivering a solution close to the reference solution. However, because the clamped plate is a more demanding benchmark, the solution is not as close as for the simply supported plate. FEM requires more nodes along the thickness to achieve a closer solution. For the RPIM (Figure 4d), both KT1 and KT-All algorithms produce solutions very close with the literature.  
Notice that in the present work, a full 3D-deformation theory is considered, which is fundamentally different from the first-order shear deformation theory used by the reference solution. The full 3D-deformation theory strongly depends on the number of nodes along the thickness. Thus, the similarity between the obtained results and the solution documented in the literature indicates that both FEM and RPIM nonlinear solution algorithms were successfully implemented.  
It would be expected that we would obtain similar results regardless of the nonlinear solution algorithm used (KT0, KT1 or KT-All). However, due to the total computational time of the analysis, per increment only a maximum of 100 iterations is allowed. The KT-All never requires more than roughly 10 iterations; however, when a large portion of integration point yield, KT1 starts to require more than 100 iterations per increment, and from third increment (very early), KT0 already requires more than 100 iterations per increment. This is because in each increment the solution is not fully corrected (the number of iterations is insufficient), the solutions start to accumulate convergence errors and, consequently, to diverge. This, observation highlights the advantage of using the full Newton-Raphson method (KT-ALL).