Deformation and internal forces of statically determinate beams using discontinuity functions: development of software

Abstract

This work presents the development of a new numerical application called "Magie", which uses mathematical discontinuity functions to calculate internal forces and generalized displacements of simply supported beams with multiple loads. Developed on the MATLAB platform, the application can calculate the bending moment M, shear force V, elastic line v(x), slopes θ(x), rotation θ, deflection v, and support reactions of simply supported, homogeneous, and isotropic beams with uniformly distributed loads, concentrated forces, and moments. Its user-friendly interface includes windows, buttons, guides, and graphs. The inputs include the geometry of the beam (positions of the supports, length, and cross-section), the applied loads, and Young's modulus (E). The numerical results obtained are validated by comparisons with exact and numerical solutions from the literature. The differential equations governing the problem are formulated in terms of discontinuity functions, reducing the number of unknowns compared to traditional solutions that use functions by beam segment. Free and in Portuguese, the application follows the national system for plotting internal force diagrams and is useful for theoretical and practical courses in engineering.