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7.9: Relation Between the Shear and Bending Moment

JoVE Core
Mechanical Engineering

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Relation Between the Shear and Bending Moment

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.

Figure 1

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.

Equation 1

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.

Equation 2

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.

Equation 3

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.

Suggested Reading


Keywords: Shear Force Bending Moment Beam Equilibrium Distributed Load Concentrated Load Couple Moment Moment Diagram Shear Diagram Structural Behavior

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