How to Get the Area and Centroid of Cantilever Moment Diagrams Easily?

Moment Diagram: Area and Centroid

Let's draw the moment diagram of our cantilever beams:

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One great thing about these moment graphs is it's easy to compute their area and centroid.

For any equation of the form \(y=kx^n\), there is a relationship among the degree (\(n\)), area (\(A\)), and centroid (\(\bar{x}\)). With basic theories on calculus, we can express it using these equations:

\(A=\frac{1}{n+1}bh\)

\(\bar{x}=\frac{1}{n+2}b\)

Example

Let's look at the table again and illustrate an example of using these equations. A concentrated load (with a 1st degree) has an area and centroid of:

\(A=\frac{1}{1+1}bh=\frac{1}{2}bh\)

\(\bar{x}=\frac{1}{1+2}b=\frac{1}{3}b\)

Summary

Let's summarize:

We can observe a striking pattern when analyzing the moment diagram of the cantilever beam.

We can observe it through a series of cantilever beams with the following loading conditions: (1) a couple at the free end, (2) concentrated point load at the free end, (3) uniform distributed load from 0 to \(L\), and (4) a uniform varying load from 0 to \(L\) (with the highest magnitude at the fixed support).

The moment equations are polynomial functions of the form \(M=kx^n\).

The moment equation's degree \(n\) is an important variable.

These moment equations give diagrams in which we can quickly compute for its area and centroid, which are \(A=\frac{1}{n+1}bh\) and \(\bar{x}=\frac{1}{n+2}b\) respectively.