# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 4 Numerical Results  
In this section, simply supported and clamped 3D square plates are analysed. The three nonlinear solution algorithms (KT-0, KT-1, and KT-ALL) are used and compared by using the RPIM and the FEM for both plate configurations. The results obtained are compared with solutions documented in the literature [26], which were obtained by using the element free Galerkin method (EFGM) and an equivalent single-layer deformation theory, the first order shear deformation theory, and matched/validated with advanced FEM formulations, equally assuming an equivalent single layered deformation theory.  
Regarding the geometric properties of the plates, following Figure 3 representation, the length \(L\) and thickness \(h\) of all analysed cases are: \(L=6.0\) m and \(h=0.2\) m. Concerning the plate's material, the following mechanical properties were considered: elasticity (Young's) modulus: \(E=206.7\) GPa, Poisson's coefficient: \(\
u=0.3\), tangent elasticity modulus: \(E_{T}=2.067\) GPa, and yield stress: \(\sigma_{Y}=206.7\) MPa.  
Regarding the loading conditions, all plates are submitted to a uniform distributed load covering the complete surface of the plate, as shown in Figure 3. In order to reduce the computational cost of the nonlinear analyses, instead computing the complete plate, only one quarter of the plate is analysed, assuming the corresponding boundary conditions. Thus, in surface \(AA^{\prime}B^{\prime}B\) the displacement on \(Ox\) direction is assumed constrained: \(\bar{u}=0\), in surface \(AA^{\prime}D^{\prime}D\) the displacement on \(Oy\) direction is assumed constrained: \(\bar{v}=0\), and in line \(AA^{\prime}\) both \(Ox\) and \(Oy\) directions are constrained: \(\bar{u}=0\wedge\bar{v}=0\). These symmetry boundary conditions are common to both simply supported and clamped plate examples. In addition, for the simply supported plate, the line of nodes on the middle layer of the plate (with \(z=0\)) have all degrees of freedom constrained: \(\bar{u}=0\wedge\bar{v}=0\wedge\bar{w}=0\). For the clamped plate, in addition to the symmetry boundary conditions, all the points belonging to surfaces \(BB^{\prime}C^{\prime}C\) and \(DD^{\prime}C^{\prime}C\) have all degrees of freedom constrained: \(\bar{u}=0\wedge\bar{v}=0\wedge\bar{w}=0\).  
All examples are analysed by using the FEM and the RPIM. Concerning FEM, eight-node hexagonal elements are considered, assuming full integration: \(2\times 2\times 2\) integration points following the Gauss-Legendre quadrature scheme. Regarding the RPIM, the background integration mesh coincides with the FEM element mesh, being formed by hexagonal integration cells filled with \(2\times 2\times 2\) integration points respecting the Gauss-Legendre quadrature scheme, and each influence domain is composed of 27 nodes. The RPI shape functions are constructed with a constant polynomial basis, \(\mathbf{p}(\mathbf{x}_{I})=1\), and the MQ-RPI shape parameters: \(c=10^{-4}\) and \(p=1-10^{-4}\).  
The problem domain is dicretized with a 3D regular nodal of of \(16\times 16\times 7=1792\) nodes, which corresponds to \(15\times 15\times 6=1350\) hexagonal elements in FEM, or hexagonal integration cells in RPIM.