Implementação computacional do método da rigidez direta para análise de pórticos planos com barras de inércia variável

Abstract

The structural analysis of plane frames is a fundamental area of structural engineering that has evolved significantly with the advent of scientific computing. This work presents the de velopment of a computational application for plane frame analysis using the Direct Stiffness Method, implemented in the Python programming language. The main objective is to create an efficient and accessible computational tool that allows structural analysis of plane frames with constant and variable cross-section elements, providing accurate results for nodal displacements, internal forces, and support reactions. The adopted methodology is based on the systematic implementation of the Direct Stiffness Method, considering the classical matrix formulation of structural analysis. The application development uses the scientific libraries NumPy and SciPy for matrix operations and linear system solving, while the graphical interface is built with the Tkinter library, providing an intuitive user experience. A distinctive feature of the application is the ability to analyze elements with variable cross-sections through the imple mentation of numerical integration techniques, significantly expanding the structural modeling possibilities. The theoretical framework covers the fundamentals of matrix structural analysis, with emphasis on the Direct Stiffness Method, including the formulation of elemental stiffness matrices, coordinate transformation procedures, global stiffness matrix assembly, and boundary condition application. The work also presents the mathematical foundation for treating variable cross-section elements, using the Flexibility Method with numerical integration. Tool validation will be performed through comparison with known analytical solutions and results obtained by established commercial software, ensuring the accuracy and reliability of the implemented calculations. It is expected that this application will contribute both to structural analysis teaching and practical applications in structural design offices, offering a free and educational alternative to existing commercial textitsoftware.