7.9 : Relation Between the Shear and Bending Moment

When a beam is subjected to various loads, such as a distributed load, concentrated loads, and a couple moment, it experiences both shear forces and bending moments. To understand the relationship between these two forces, we can analyze an elemental section of the beam and draw a free-body diagram.

For the elemental section of the beam to be in equilibrium, the moment acting on the right side of the section should be higher by a small and finite amount compared to the left side. The distributed load exerts a resultant force at a fractional distance from the section's right end. We can use the equilibrium equation for moment to establish the relationship between the shear and bending moment.

By dividing this equation by Δx and letting Δx approach zero, we determine the slope of the moment diagram, which is equivalent to the shear.

By integrating the distributed load over the elemental section lying between two arbitrary points, we can correlate the change in the bending moment and the area under the shear diagram.

This relationship is essential for understanding how the beam's internal forces respond to external loads and how these forces impact the beam's overall structural behavior.