To construct a bending moment diagram for an overhanging beam, start by drawing the free-body diagram and identifying all forces acting on the beam. If the upward forces are equal, treat them as a uniformly distributed load (UDL); if they vary, consider them as a varying distributed load (UVL). For a semi-elliptical load distribution, approximating the load as point loads can simplify calculations. The bending moment can be calculated using the formula F1 * d + F2 * 2d, where F represents the forces and d the distances from the point of interest. Understanding these concepts is crucial for accurately analyzing the bending moments in structural engineering applications.