# Two-dimensional nonlinear analysis of elastic columns stability using the convected particle domain interpolation material point method

## Palavras-chave:

Buckling, Columns, Convected particle domain interpolation, Material point method, Stability## Resumo

This paper presents an elastic two-dimensional analysis of columns stability, using an extension of the

material point method (MPM), called convected particle domain interpolation (CPDI). As instability phenomena

require a large-deformation study, MPM is chosen by its ability to deal with this situation, using a simple and reg-

ular cartesian background grid to calculate the spatial gradients and divergences, besides automatically considers

explicit dynamics. The CPDI extension is applied in order to facilitate natural boundary conditions treatment and

to avoid the cell-crossing error, since the subdomains are explicitly tracked through the analysis and the shape func-

tions have continuous derivatives into these subdomains. A computational routine is implemented using Python

3 language, employing Euler-Gauss explicit time integration and update-stress-last (USL) scheme. A rectangular-

section slender column is submitted to 4 different load cases. The results obtained are the load-displacement

behavior of the columns and its total mechanical energy time variation, which are validated by comparison with

the results produced by classic analytical Euler’s theory. It can be verified, after the analyses, that CPDI MPM is

able to predict the columns buckling loads and to reproduce the post-critical instability phenomena with a good

accuracy. The conservation of energy is achieved in all tests.