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 b…
Suppose a beam is subjected to a distributed load, two concentrated loads, and a couple moment. Establish the relationship between the shear and the bending moment.
Consider an elemental section of the beam and draw a free-body diagram of the section.
For the section to be in equilibrium, the moment acting on the right side of the section should be higher by a small and finite amount.
A resultant force of the distributed load is exerted at a fractional distance from the section's right end.
Using the equilibrium equation for moment, a relation between moment and shear can be obtained.
Further, by dividing it by Δx and letting Δx approach zero, the slope of the moment diagram is determined, which is equivalent to the shear.
A maximum bending moment occurs at the point where the slope of the moment and the shear are zero.
Integrating the distributed load over the elemental section, lying between two arbitrary points, a correlation between the change in the bending moment and the area under the shear diagram is obtained.
View the full transcript and gain access to JoVE Core videos
Q1: How do you establish the relationship between shear and bending moment in a beam?
To establish the relationship, analyze an elemental section of the beam using a free-body diagram. Apply equilibrium equations for moments, noting that the moment on the right side should be higher by a small amount. By dividing the moment equation by Δx and letting Δx approach zero, you determine the slope of the moment diagram, which equals the shear force.
Q2: What does the slope of the moment diagram represent?
The slope of the moment diagram is equivalent to the shear force acting on the beam. This relationship is derived by dividing the moment equilibrium equation by the elemental length Δx and taking the limit as Δx approaches zero, establishing a direct mathematical connection between moment and shear.
Q3: Where does the maximum bending moment occur on a beam?
The maximum bending moment occurs at the point where both the slope of the moment diagram and the shear force are zero. This critical location represents a transition point where the shear changes sign, making it essential for structural design and analysis.
Q4: How does the area under the shear diagram relate to bending moment?
By integrating the distributed load over an elemental section between two arbitrary points, a correlation emerges: the change in bending moment equals the area under the shear diagram. This relationship provides a graphical method for determining moment variations along the beam's length.
Q5: Why must the moment on the right side of an elemental section be higher than on the left?
For the elemental section to remain in equilibrium, the moment on the right side must be higher by a small finite amount to balance the effects of shear forces and distributed loads acting on the section. This moment difference is essential for maintaining rotational equilibrium of the infinitesimal element.
Q6: What role does the distributed load play in the shear-moment relationship?
The distributed load exerts a resultant force at a fractional distance from the section's right end, creating the moment differential needed for equilibrium. When integrated over the elemental section, the distributed load determines how shear and moment vary along the beam, directly influencing the shape of both diagrams.
Q7: How do concentrated loads and couple moments affect the shear-moment relationship?
Concentrated loads and couple moments contribute to the overall loading condition on the beam, affecting both shear and bending moment distributions. These loads must be included in the free-body diagram analysis to accurately establish the relationship between shear and moment throughout the beam.