Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

Consider a beam experiencing a distributed load, two concentrated loads, and a couple moment. Establish the relationship between the shear and the distributed load.
First, consider an elemental section on the beam free from any concentrated load or couple moment and draw a free-body diagram of the section.
To maintain equilibrium, the shear force acting on the right-hand side of the section should be incremented by a small and finite amount.
The resultant force of the distributed load acts at a fractional distance from the right end of the section.
Using the equation of equilibrium for vertical force, a relation between shear and load is obtained.
Next, by dividing both sides of the equation by &Delta;x and letting &Delta;x approach zero, the slope of the shear force can be determined, which is equal to the distributed load intensity.
Finally, rearranging the equation and integrating the distributed load over the elemental section between two arbitrary points leads to a relation between the change in shear and the area under the load curve.

relation between the distributed load and shear

Relation Between the Distributed Load and Shear

Understanding the relationship between the distributed load and shear force in structural analysis is crucial for analyzing beams subjected to various loading conditions. Consider the case of a beam experiencing a distributed load, two concentrated loads, and a couple moment.
<img alt="Static equilibrium diagram with forces F1, F2; equations ΔF=w(x)Δx, Δk=k(Δx); mechanical setup." src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium diagram with forces F1, F2; equations ΔF=w(x)Δx, Δk=k(Δx); spring load setup.">
The connection between the shear force and the distributed load for the given case can be established following the given procedure. First, consider an elemental section on the beam free from any concentrated load or couple moment. We can draw a free-body diagram to analyze the forces acting on this section. The diagram will consist of the distributed load acting along the length of the beam, the shear force V(x) acting on the right-hand side of the section, and an incremental shear force dV added to maintain the equilibrium.
To maintain equilibrium in the vertical direction, the shear force acting on the right-hand side of the section should be incremented by a small and finite amount, ΔV. The resultant force of the distributed load, w(x)Δx, acts at a fractional distance from the right end of the section.
Using the equation of equilibrium for vertical force, the following relation between shear and load can be obtained:
<img alt="Static equilibrium equation, ΔV=-w(x)Δx, formula in mechanics diagram for load distribution analysis." src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;" data-altupdatedat="1754904545000" data-alt="Change in potential energy formula, ΔV=-w(x)Δx, static equilibrium, concept diagram.">
Next, we will divide both sides of the equation by Δx and let Δx approach zero:
<img alt="Static equilibrium formula, \( \frac{dV}{dx} = -w(x) \), mathematical equation." src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation \( \frac{dV}{dx} = -w(x) \), formula for load distribution analysis.">
This equation shows that the slope of the shear force is equal to the distributed load intensity.
Finally, we can rearrange the equation and integrate the distributed load over the elemental section between two arbitrary points, Q and R:
<img alt="Static equilibrium equation; VR-VQ=−∫w(x)dx diagram. Concepts: force balance, integral calculation." src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation: VR-VQ=-∫w(x)dx, integral formula for physics, mathematics.">
This integral equation demonstrates the relationship between the change in shear force and the area under the load curve. The difference in shear force between points Q and R equals the area under the distributed load curve between these two points.

Understanding the relationship between the distributed load and shear force in structural analysis is crucial for analyzing beams subjected to various ...

relation-between-the-distributed-load-and-shear

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shear change under distributed loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment.

To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

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Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

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key terms and concepts

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | JoVE K12

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Shear Change Under Distributed Loading

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