In beam theory, the shear force and bending moment at a cross-section vary along the beam's length according to differential relationships with the applied transverse loading: the rate of change of shear force with position equals the load intensity, and the rate of change of bending moment with position equals the negative of the shear force. These relationships, derived from equilibrium of an infinitesimal beam element, describe how distributed loads, concentrated forces, and concentrated moments each produce distinct signatures in the shear and moment fields — constant shear under no load, linear shear under uniform load, an abrupt jump in shear at a concentrated force, and an abrupt jump in moment at a concentrated moment with no accompanying jump in shear. This is a core topic within mechanics of materials (strength of materials), specifically the analysis of beams under transverse loading, building on the prior definitions of shear force and bending moment as internal stress resultants.
In beam theory, the shear force and bending moment at a cross-section vary along the beam's length according to differential relationships with the applied transverse loading: the rate of change of s…