This paper presents a comprehensive investigation into the topology optimisation of steel truss bridge structures using gradient-based mathematical algorithms, with particular emphasis on the Solid Isotropic Material with Penalisation (SIMP) method and the Method of Moving Asymptotes (MMA). The structural compliance minimisation problem is formulated within a continuum mechanics framework, subject to volume fraction constraints and equilibrium conditions derived from the finite element method (FEM). A sensitivity analysis scheme employing adjoint variables is implemented to compute the gradient of the objective function with respect to element density design variables. The effect of density filtering, projection techniques, and penalisation parameters on convergence behaviour and topological clarity of the optimised designs is systematically investigated. Numerical experiments conducted on a simply-supported 100 m steel truss bridge demonstrate that topology optimisation achieves a material saving of up to 36.8% compared with conventional design, while maintaining full compliance with Eurocode 3 and AASHTO load criteria. Results confirm that gradient-based methods converge reliably within 60-80 iterations for mesh sizes up to 200 x 100 elements. The optimised topologies exhibit characteristic diagonal-member configurations that are consistent with classical Michell truss theory. This study contributes a verified computational methodology applicable to large-span bridge engineering practice in Sub-Saharan Africa and beyond.