Relation Between the Distributed Load and Shear

The distributed load directly determines the slope of the shear force diagram. When you divide the change in shear force by the elemental length and let that length approach zero, the slope of the shear force equals the distributed load intensity at that point. This mathematical relationship shows that steeper load distributions create steeper changes in shear force along the beam.

The free-body diagram of an elemental beam section shows that the shear force must increase by a small amount ΔV on the right side to balance the distributed load acting downward. This increment maintains vertical equilibrium. The resultant force of the distributed load acts at a fractional distance from the right end, creating the need for this shear force adjustment.

The change in shear force between any two points on a beam equals the area under the distributed load curve between those points. By integrating the distributed load over an elemental section, you obtain this integral relationship. This principle allows engineers to determine shear force changes without calculating forces at every location along the beam.

The slope relationship emerges from the equilibrium equation for vertical forces on an elemental section. When you rearrange the equilibrium equation and divide both sides by the elemental length Δx, then let Δx approach zero, you obtain dV/dx = -w(x). This derivative shows that the rate of change of shear force with respect to position equals the negative distributed load intensity.

Start by isolating an elemental beam section free from concentrated loads or couple moments. Draw a free-body diagram showing the distributed load, shear force V(x) on the right side, and incremental shear force ΔV. Apply vertical force equilibrium: the sum of forces must equal zero. This equilibrium condition directly yields the mathematical relationship between shear and distributed load.

The resultant force of the distributed load, calculated as w(x)Δx, acts at a fractional distance from the right end of the elemental section. This resultant creates the vertical force that must be balanced by the increment in shear force. Understanding where this resultant acts is essential for correctly applying equilibrium equations and determining shear force changes.

Integrating the distributed load intensity over an elemental section between two arbitrary points yields the total change in shear force. This integral relationship demonstrates that the area under the load curve directly equals the difference in shear force values at those two points. Integration transforms the local slope relationship into a global change relationship useful for shear and bending moment diagram problem solving.