Deformed shape of 2D frames composed of parabolic arc-shaped and straight elements using differential equations.

Resumo

This paper presents the differential relations for displacements, and rotations of parabolic arc-shaped elements subjected to various loads, along with their computational implementation in a plotting algorithm for 2D frames composed of both curved and straight bars. The program is written in Python and is based on the Finite Element Method. The differential equations for straight elements, which are well-established, provide the basis for formulating those applicable to the parabolic arcs. The stiffness matrices are obtained by inverting the flexibility matrices, which are calculated using the Force Method along with a Gauss-Legendre Quadrature applied to evaluate specific integrals. The program is validated against several frame structures analyzed using commercial software, in which the arc segments are approximated by a series of straight elements. Results for displacements and rotations are shown to agree very well with those calculated using the Virtual Work Method, whereas the results from commercial software vary significantly with the level of discretization.