7.9:
Relation Between the Shear and Bending Moment
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.
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.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 370-371.
- Beer, F.P.; Johnston, E.R.; Mazurek, D.F; Cromwell, P.J. and Self, B.P.(2019). Vector Mechanics for Engineers ‒ Statics and Dynamics. New York: McGraw-Hill. Pp 391-392.