# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 4 Numerical Results
### Elastoplastic Analysis of 3D Structures
#### 4.2.1 Two-Bay Asymmetric Frame  
The same dimensions used in the 2D case are applied here as well. The thickness variation for the 3D model is represented in Figure 19, as an upper view of the model. Only half of the structure is analyzed due to symmetry. As for nodal density, a discretization of 2481 nodes was used.  
Additionally, due to symmetry, the \(w\) displacement is locked, which is also represented in Figure 19. Similarly with the 2D case, for the 3D case, the horizontal displacement of point **A** is presented in Figure 20, and it is compared with the reference solution. The units used are the same as in the 2D case. As it can be seen in the referred figure, although not as coincident with the reference solution given by [PERSON] et al. when compared with the 2D solutions, the obtained results are nonetheless similar to the reference values. This deviation is mainly caused by the nodal density used. For 3D geometries, the computational cost of a high nodal density, considering the computational power that is available, was very high. Therefore, considering the time that it would take for the calculations to be complete, the nodal density used was considered satisfactory.  
Figure 19: Thickness variation and additional boundary condition for the 3D model—Upper view.  
The effective plastic strain distribution maps for the FEM, RPIM, and NNRPIM, considering the 3D case, are presented in Figures 21-23.  
Figure 20: Horizontal displacement of point **A** given by the FEM (**a**), the RPIM (**b**), and the NNRPIM (**c**). The results from [PERSON] et al. can be found in [33].  
Figure 22: Development of the plastic connections for the RPIM—3D case: (**a**) 19 \(\frac{kilopound}{10}\). (**b**) 25 \(\frac{kilopound}{10}\). (**c**) 30 \(\frac{kilopound}{10}\). (**d**) 42 \(\frac{kilopound}{10}\).  
Analyzing Figures 21-23 and comparing them to the 2D case, the same plastic connections are developed. In the 3D case, these plastic connections are more broad, meaning that the plastic strain on these critical points is more severe.  
In terms of the progression of the plastic connections, as the load increases, for all three numerical methods, we can see the same evolution. An interesting detail to point out is the absence of plastic connection 6 in Figure 21d. The plastic connection is not shown due to the fact that the load increment after \(36\xrightarrow{kilopound}\frac{}{10}\) leads to the complete hardening of the material, and due to that fact, the plastic connection is not evident. However, as it is shown in Figure 20a, this plastic connection is developed but at a lower load. Overall, the results are in agreement, as they should be, with the 2D case. The diagrams showing the evolution of the effective stress on the identified plastic connections are presented in Figure 24. The results obtained by the three methods are similar. The same detail presented for the 2D case, concerning the entrance of plastic connection 2 in the elastoplastic regime at first, with plastic connection 1 having a rapid transition between elastic and plastic behavior, is also evident. The simultaneous entrance into plasticity of plastic connections 3 and 4 is also evident since both curves shift at the same time.  
Figure 24: Evolution of the effective stress on the plastic connections given by the FEM (**a**), the RPIM (**b**), and the NNRPIM (**c**)—3D case.  
The results presented for the horizontal displacement of point **A** for the 2D and 3D case will now be presented in a combined diagram, in Figure 25, for a more complete comparison.  
The combined results shown in Figure 25 show good agreement between the 2D and 3D results, for all three numerical methods, and a combined agreement with the reference solution by [PERSON] et al. [33].