# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Elastoplastic Analysis of 2D Frames
#### 4.1.1 Two-Bay Asymmetric Frame  
The analyzed frame has its basis in the research paper by [PERSON] et al. [33]. For this example, a nodal discretization of 6063 nodes was used for the FEM, RPIM, and NNRPIM. The geometry and dimensions of the two-bay asymmetric frame considered are represented in Figure 7. The mechanical properties of the material are given by Table 2.  
The base of elements 1 and 5 are completely locked in all degrees of freedom (\(u\) and \(v\)), while a connection at the base of element 4 is considered (\(R\)), allowing the rotation of the element, with the displacements being locked as well. The length of the elements considered and the cross-section dimensions are shown in Figure 7.  
The horizontal displacement graphs of point \(A\), presented in Figure 7, are now shown in Figure 8. As for units, the \(x\) axis is normalized by the relationship \(\frac{u_{A}}{6}\), and the \(y\) axis is correspondent to the load in \(\frac{kilopound}{10}\). As it can be seen in Figure 8, the results obtained coincide well with the reference solution for all three numerical methods. As for the marks labeled with respective numbers, these concern the plastic connections developed as the load increases. These plastic connections are represented in Figure 9. Comparing the load values for the six plastic connections calculated using the meshless methods with the FEM solutions, it is possible to obtain the percentage errors shown in Table 3. As expressed in this table, the relative difference between the RPIM results and the FEM is always lower than 6.73%, while in the case of the NNRPIM, the difference is lower than 4.07%. This quantitative comparison highlights the accuracy of the meshless methods studied in this work. The same tendency is verified in the results presented in the coming subsections. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mechanical Properties** & **Value** \\ \hline \(E\) [Pa] & 19,613.3 \(\times\) 10\({}^{6}\) \\ \(E_{T}\) [Pa] & 19,613.3 \(\times\) 10\({}^{2}\) \\ \(\
u\) & 0.3 \\ \(\sigma_{Y}\) [Pa] & 98.0665 \(\times\) 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of the two-bay asymmetric frame. Figure 7: Geometry, boundary, and load conditions of the two-bay asymmetric frame. Figure 8: Horizontal displacement of point **A** given by the FEM, RPIM, and NNRPIM. The results from [PERSON] et al. can be found in [33]. The effective plastic strain distribution maps for the FEM, RPIM, and NNRPIM are presented in Figures 10-12. As it can be seen, as the load increases, the amount of effective plastic strain at a certain point of the structure increases, creating plastic connections or plastic hinges. These connections are created due to the plastification of certain points in the structure that are subjected to high stress considering the load input. This way, we can identify the critical points of the structure. Comparing Figures 10-12, the three methods return the same critical points, with the same development of the plastic connections between them. The diagrams showing the evolution of the effective stress on the identified plastic connections are presented in Figure 13. As can be seen in the figure, for all three numerical methods used, the evolution of the effective stress is quite similar. An interesting detail to be analyzed is the evolution of plastic connections 1 and 2. In fact, plastic connection 2 enters in plastic regime first, when compared to plastic connection 1. Comparing Figures 10-12, in terms of effective plastic strain, both data seem to contradict each other. However, in truth, plastic connection 1 has a very rapid transition between elastic and plastic behavior, while plastic connection 2 is slower in the same transition, as can be assessed by the curvature of both plastic connections shown in the diagrams. Another important aspect of the diagrams presented is the stabilization of the stress level at a certain load applied.