What are Load, Shear, and Moment Relationships?

There is a relationship among the external loads, shear, and moment that is extremely useful, especially when creating shear and moment diagrams without the equations.

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There is a relationship among the external loads, shear, and moment that is extremely useful:

Load, Shear, and Moment Relationships

Say we have a simple beam with a random load $w$ . Let's consider a cut section between two points $x_{1}$ and $x_{2}$ along the structure with a minuscule length of $d x$ .

Let's discuss the forces of this free-body diagram when we cut this section:

We can think of the external load $w$ as a uniform distributed load with a magnitude of $w \times d x$ .
We exposed shear $V$ and moment $M$ at both sides.
On the left, we have shear $V$ and moment $M$ . On the right, we have shear $V + d V$ and moment $M + d M$ .

Let's apply the equilibrium equations for this section. Starting with vertical equilibrium:

$\sum F_{v} = 0] ↑_{+}$

$V - w (d x) - (V + d V) = 0$

Equation 1: $d V = w (d x)$

Equation 2: $w = \frac{d V}{d x}$

Next, we consider taking a summation of moments at point $x_{2}$ :

$\sum M_{x_{2}} = 0] ↻_{+}$

$M + V (d x) - (w \times d x) \frac{d x}{2} - (M + d M) = 0$

The square of the differential $d x^{2}$ is negligible, hence, we have:

Equation 3: $d M = V (d x)$

Equation 4: $V = \frac{d M}{d x}$

With these four equations, we're ready to discuss the relationships between load, shear, and moment.

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Relationship Between Load and Shear

Let's consider Equations 1 and 2 and find relationships between load and shear:

Change in Shear

Say we integrate Equation 1:

$\int_{V_{1}}^{V_{2}} d V = \int_{x_{1}}^{x_{2}} w \times d x$

Change in Shear: $V_{2} - V_{1} = Δ V = \int_{x_{1}}^{x_{2}} w \times d x$

This equation is one key relationship between load and shear. It means that if you take the area of the load diagram between positions $x_{1}$ and $x_{2}$ , you'll get the change in shear (difference of shear values) between the said two positions.

Shear Slope

We can find another key relationship with Equation 2:

Shear Slope: $w = \frac{d V}{d x}$

It states that the slope of the shear equation $\frac{d V}{d x} = V^{'}$ at a specific point is equal to the value of the load $w$ at said point.

Relationship Between Shear and Moment

Let's consider Equations 3 and 4 and find relationships between shear and moment:

Change in Moment

Say we integrate Equation 3:

$\int_{M_{1}}^{M_{2}} d M = \int_{x_{1}}^{x_{2}} V \times d x$

Change in Moment: $M_{2} - M_{1} = Δ M = \int_{x_{1}}^{x_{2}} V \times d x$

This equation is one key relationship between shear and moment. It means that if you take the area of the shear diagram between positions $x_{1}$ and $x_{2}$ , you'll get the change in moment (difference of moment values) between the said two positions.

Moment Slope

We can find another key relationship with Equation 4:

Moment Slope: $V = \frac{d M}{d x}$

It states that the slope of the moment equation $\frac{d M}{d x} = M^{'}$ at a specific point is equal to the value of the shear $V$ at said point.

Shear and Moment Diagrams

These four fundamental relationships tell us that load, shear, and moment are interconnected. This implies that we can construct shear and moment diagrams from these relationships without using shear and moment equations.

Application

Shear diagram using load-shear relationships
Moment diagram using shear-moment relationships

Summary

Let's summarize:

There is a relationship among the external loads, shear, and moment that is extremely useful, especially when creating shear and moment diagrams without the equations.

Change in Shear. $V_{2} - V_{1} = Δ V = \int_{x_{1}}^{x_{2}} w \times d x$ . If you take the area of the load diagram between positions $x_{1}$ and $x_{2}$ , you'll get the change in shear (difference of shear values) between the said two positions.

Shear Slope. $w = \frac{d V}{d x}$ . The slope of the shear equation $\frac{d V}{d x} = V^{'}$ at a specific point is equal to the value of the load $w$ at said point.

Change in Moment. $\int_{M_{1}}^{M_{2}} d M = \int_{x_{1}}^{x_{2}} V \times d x$ . If you take the area of the shear diagram between positions $x_{1}$ and $x_{2}$ , you'll get the change in moment (difference of moment values) between the said two positions.

Moment Slope. $V = \frac{d M}{d x}$ . The slope of the moment equation $\frac{d M}{d x} = M^{'}$ at a specific point is equal to the value of the shear $V$ at said point.

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Created On

June 5, 2023

Updated On

February 23, 2024

Contributors

Edgar Christian Dirige

Founder

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