7.8 : Relation Between the Distributed Load and Shear

Understanding the relationship between the distributed load and shear force in structural analysis is crucial for analyzing beams subjected to various loading conditions. Consider the case of a beam experiencing a distributed load, two concentrated loads, and a couple moment.

The connection between the shear force and the distributed load for the given case can be established following the given procedure. First, consider an elemental section on the beam free from any concentrated load or couple moment. We can draw a free-body diagram to analyze the forces acting on this section. The diagram will consist of the distributed load acting along the length of the beam, the shear force V(x) acting on the right-hand side of the section, and an incremental shear force dV added to maintain the equilibrium.

To maintain equilibrium in the vertical direction, the shear force acting on the right-hand side of the section should be incremented by a small and finite amount, ΔV. The resultant force of the distributed load, w(x)Δx, acts at a fractional distance from the right end of the section.

Using the equation of equilibrium for vertical force, the following relation between shear and load can be obtained:

Next, we will divide both sides of the equation by Δx and let Δx approach zero:

This equation shows that the slope of the shear force is equal to the distributed load intensity.

Finally, we can rearrange the equation and integrate the distributed load over the elemental section between two arbitrary points, Q and R:

This integral equation demonstrates the relationship between the change in shear force and the area under the load curve. The difference in shear force between points Q and R equals the area under the distributed load curve between these two points.