This post illustrates how to construct the moment diagram of a beam structure using Moment By Parts. We shall explore how making this diagram differs from a usual moment diagram.

The solution presented is in SI. The author will update the post soon to reflect English units.

Main Solution

The following is a general outline of how to use moment-by-cantilever parts:

Strategically Cut the Beam Into Parts

The first step is to cut the beam into two parts - the left and right sections. For this example, say we cut it at point \(C\). We'll discuss soon why we put it at \(C\).

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After dividing the beam into two parts, we isolate each loading condition in each section.

For section \(ABC\), there are three loading conditions; hence, we break down this section into three cantilever beams with the following loads:

\(27kN•m\) clockwise couple at \(A\)

\(90kN\) upward reaction at \(B\)

\(0\rightarrow{45kN/m}\) downward uniform varying load between \(B\) and \(C\)

For section \(CDE\), there are also three loading conditions; hence, we break down this section into three cantilever beams with the following loads:

\(36kN/m\) downward uniform distributed load between \(C\) and \(D\)

\(263.25kN\) upward reaction at \(D\)

\(90kN\) downward point load at \(E\)

With six cantilever beams, we create the moment diagram for each using cantilever beam patterns. As a result, we have the following graphs:

Superimpose Each Part

After drawing the moment diagrams for each part, we combine all of these graphs into one single graph:

Moment Diagram (By Parts)

This graph is the moment diagram of the beam after superimposing all moment diagrams. The benefit of this moment-by-parts diagram is that it makes it easy for us to compute the area and centroids of each part - something we would need when calculating the beam's deflection.

Traditional Moment Diagram and Superimposed Moment Diagram

You might be thinking: "how is this moment diagram the same as when we create moment diagrams using equations?"

Look at the moment-by-parts diagram and consider the moment values at \(C\). If we add all the moments at this location (on either side):

\(\sum M_{C_L}=27+405-151.875=280.125\)

\(\sum M_{C_R}=-364.5+1184.625-540=280.125\)

This superimposed value is the same as if we find the moment at \(C\) using the corresponding moment equation:

\(M_{B C}=-\frac{5}{3} x^3+10 x^2+70 x-139 \frac{2}{3}\{2 \leq x \leq 6.5\}\)

This equality implies that if you add the corresponding moment values in the parts diagram, you will eventually arrive at the moment graph made using equations.

Moment Diagram

Why There?

Let's go back to why we chose point \(C\) as the location of the cutting plane. In actuality, we can select any location to place it; however, we are taking advantage of the cantilever beam patterns for an easy solution. So, we need to keep these points in mind when choosing a place:

We need to mimic the loading conditions of the cantilever beam patterns. If, for instance, we place it somewhere between \(B\) and \(C\), we will complicate the loading conditions on the right side of the beam.

We should minimize the number of loading conditions to break down when making cantilever parts. Again, if we place it between \(B\) and \(C\), the number of cantilever beams on the right side may require more than three.

The goal of the moment-by-parts diagram is to make it easy for us to compute the area and centroid of the moment graph (which we will need in specific topics). The patterns help us in this case. Hence, we should place it so that the resulting diagram would be easy to compute for its area and centroid.