# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 1 Introduction  
In Computational Mechanics, the Finite Element Method (FEM) is a standard numerical tool capable of analyzing several problems. However, when some of those problems involve mesh distortions, the FEM may be not the most suitable method [1]. Meshless methods are an alternative to standard methods like the FEM. In these methods, the domain is discretized by a random set of nodes rather than an element mesh. Then, the influence-domain concept is applied instead of an element concept, so the nodes can be distributed in a random fashion [2; 3]. These and other characteristics make meshless methods ideal for computing problems involving complex and transitory geometries [4] and non-linear processes involving the iterative use of strain and stress fields [5]. In this work, such problems are studied: the elastoplastic behavior of frames and structures. The mechanical behavior of structures is often analyzed using a linear analysis. However, the existence of non-linear phenomena applied to said structures during their working conditions is a reality, and they need to be accounted for. Non-linear behavior can bring certain economic and safety benefits to structural projects, given that after the material surpasses its yield limit, it is not necessarily true that the structure composed of said material is on its way to collapse. In reality, that same structure can still withstand loading. Accounting for elastoplastic behavior leads to a more complete project, not only in safety terms but also in economic terms, with dimensions of profiles and structural elements being inferior to the ones used in all-elastic domain design. The elastoplastic theory is a mandatory subject in the current need for thinner, lighter, and more mechanically resistantstructural elements since accounting for the elastoplastic effects is a way to reduce costs and increase the safety of structural elements. As previously stated, this work will use meshless methods to simulate the elastoplastic behavior of several structures. Meshless methods first appeared with the Smooth Particle Hydrodynamics Method (SPH) in 1977 [6], applied to solid mechanics by [PERSON] and [PERSON] [7] only in 1991. The SPH uses a strong formulation of the physical problem in order to solve it. On the other hand, the Element-Free Galerkin Method (EFGM) [1] is the first to depend on a global weak form. Some other mesh-free methods appeared at the same time, like the Reproducing Kernel Particle Method (RKPM) [8] or the Meshless Local Petrov-Galerkin method [9]. The Natural Element Method (NEM) [10] was the first proposed method to solve the lack of the delta Kronecker property. Over the course of the next ten years, other methods appeared such as the Method of Finite Spheres (MFS) [11], the Point Interpolation Method (PIM) [12], the Point Assembly Method (PAM) [13], the Meshless Finite Element Method (MFEM) [14], the Natural Neighbor Radial Point Interpolation Method (NNRPIM) [15] and the Radial Point Interpolation Method (RPIM) [16]. The Radial Point Interpolation Method (RPIM) [16] and The Natural Neighbor Radial Point Interpolation Method (NNRPIM) [5] are used in this work. Meshless methods have been applied in some elastoplastic simulations of structures. [PERSON] et al. [17] used a centroid-enriched edge-based/face-based smoothed RPIM (CE-ES-RPIM and CE-FS-RPIM) to analyze elastoplastic benchmark problems in solid mechanics; the Meshless Local Petrov-Galerkin method was applied in the thermo-elastoplastic analysis of thick functionally graded (FG) plates by [PERSON] et al. [18]; the same author studied, for the first time, three-dimensional (3D) thermo-elastoplastic bending analysis of functionally graded sandwich plates subjected to combined thermal and mechanical loads, using a local radial point interpolation method (LRPIM) [19]; the RRKPM was used in the elastoplastic analysis of functionally graded materials by [PERSON] et al. [20]; the same meshless method was applied in the elastoplastic analyses of several solid mechanics problems by [PERSON] et al.