Differential Relations for Deflected Shape and Internal Forces in 2D Frames with Straight and Curved Elements

Resumo

This paper presents the differential relations and the development of an algorithm that calculates and plots the internal forces and deflected shape of frames with both circumferential arc-shaped and straight elements, subjected to various types of loads under linear-elastic conditions. The program is developed in Python and is based on the Finite Element Method. Differential equations describing bending, shear, and axial forces, as well as displacements and rotations along the structure, are presented. The stiffness matrices and load vectors are obtained by inverting the corresponding flexibility matrices, which are calculated using the Force Method. A Gauss-Legendre Quadrature is employed to evaluate some integrals. An example is provided, calculated and compared with results from commercial software, where curved bars must be discretized into multiple elements. Internal forces, displacements and rotations calculated by the algorithm are shown to agree very well, and in most cases, are closer to that calculated with the Virtual Work Method, while requiring less demanding input data.