Name: shear and moment diagrams in beam loads
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 22 s
Channel: JoVE

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&#39;s right end.
Using the equilibrium equation for moment, a relation between moment and shear can be obtained.
Further, by dividing it by &Delta;x and letting &Delta;x 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.

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.
<img alt="Statics equilibrium diagram with forces and equations (ΔF=w(x)Δx) showing weight distribution." src="/files/ftp_upload/14305/14305_Figure_1v2.svg" style="text-align: center; height: 350px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium diagram with force and moment analysis; ΔF=w(x)Δx, Δk=k(Δx), 0&lt;k&lt;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.
<img alt="Equilibrium equation ΔM=VΔx-kw(x)Δx² for static analysis, mathematical formula." src="/files/ftp_upload/14305/14305_Equation_1v2.svg" style="text-align: center; height: 15px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation ΔM=VΔx−kw(x)Δx²; mechanical analysis formula.">
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.
<img alt="Differential equation dM/dx=V, representing momentum change rate; mathematical concept." src="/files/ftp_upload/14305/14305_Equation_2v2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1754904545000" data-alt="Differential equation, dM/dx = V, illustrating static equilibrium concept.">
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.
<img alt="Static equilibrium equation, \( M_C - M_B = \int_{x_B}^{x_C} V \, dx \), integration diagram." src="/files/ftp_upload/14305/14305_Equation_3v2.svg" style="text-align: center; height: 45px;" data-altupdatedat="1754904545000" data-alt="Beams bending moment equation: \(M_C - M_B = \int_{x_B}^{x_C} V dx\), engineering 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.

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 ...

relation-between-the-shear-and-bending-moment

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shear and moment diagrams in beam loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments.

The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a fractional distance from the right end of the section.

Using the moment equilibrium equation, the link between shear and bending moment can be found. Dividing this equation by Δx and letting Δx approach zero gives the slope of the moment diagram. That slope is the shear.

The same idea also connects the change in bending moment to the area under the shear diagram. By integrating the distributed load over the elemental section between two arbitrary points, the change in moment can be related to the shear diagram area. This relationship is important for understanding how internal forces respond to external loads and how they affect the beam's structural behavior.

Shear and Moment Diagrams in Beam Loads

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Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

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key terms and concepts

Learn how distributed loads, concentrated loads, and couple moments create shear and bending moment in a beam, and how the diagrams are connected.

Shear and Moment Diagrams in Beam Loads | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Shear and Moment Diagrams in Beam Loads

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