Computing Internal Member Forces in a Bridge Truss Using Classical Iterative Numerical Methods with Maple® & MATLAB®

Abstract

In this study, computation and analysis of internal member forces acting on a bridge truss were carried out. First, the forces were resolved at each joint and a system of equations was built to describe the truss as a Linear System of Algebraic Equations (LSAEs). The LSAEs developed here is of the order 8 x 8 and sparse. Aside from the truss system being a sparse matrix, it is neither positive definite nor a tridiagonal matrix. Hence, a weakly diagonally dominant matrix characterised by ρ (A) > 1. Secondly, 3 iterative numerical methods were applied to obtain a solution to the LSAEs. Third, with Maple®, Jacobi and Gauss-Seidel methods were used with relative ease to the LSAEs, and its solution converged after 30 and 18 iterations respectively. When Successive Over-relaxation (SOR) method was applied with ω = 1.25, a solution to the LSAEs failed to converge. In a novel approach, the error evolution was simulated against iteration number for ω = 0.1 - 0.99 in Maple®. After analysing such results, ω = 0.93 was selected as the optimal value for the Relaxation Technique and solution to the LSAEs converged after ten iterations. MATLAB® codes were then written for the three iterative numerical methods to validate the results obtained in Maple®. The method proposed here proved to be very effective.