MATRIX ANALYSIS OF PLANAR STRUCTURES BY THE DIRECT STIFFNESS METHOD: THEORY AND COMPUTATIONAL IMPLEMENTATION

Resumo

Matrix analysis is a widely used tool for understanding the behaviour of structures when subjected to different loads. In order to gain deeper insight into matrix structural analysis and to apply the programming and structural engineering knowledge acquired, this work was developed to implement the Direct Stiffness Method computationally for solving beams, planar trusses and planar frames in both statically determinate and indeterminate systems. This study is warranted by its contribution to teaching structural analysis, providing students with an analytical tool in a field of utmost significance for civil-engineering education. The developed algorithm generates reports containing nodal displacements, internal member forces and support reactions, and ultimately produces a plot of the structure’s deformed configuration. The software used was GNU Octave®, an open-source environment offering excellent routines for solving matrices and linear systems with high precision, featuring a language compatible with MATLAB®. Five examples were analysed: a hyperstatic beam, a truss, a composite frame, a frame with a tension rod and a three-hinged frame, all drawn from well-established references in structural engineering. The results obtained were promising, since the internal forces and support reactions showed no differences when compared to those from Ftool®, with only minor discrepancies in certain displacements due to the program’s numerical precision versus that of the software; some displacements that theoretically should be zero resulted in relative zeros in both the program and the software, considering a precision of 19 decimal places. Thus, the developed program met its objectives, demonstrating effectiveness in solving planar structures.