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How to Calculate Bending Moment Diagrams?

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Bending Moment Diagram Calculation

Below are simple instructions on how to calculate the bending moment diagram of a simply supported beam. Study this method as it is very versatile (and can be adapted to many different types of problems. The ability to calculate the moment of a beam is a very common practice for structural engineers and often comes up in college and high school exams. It’s good to note early, that the SkyCiv Beam software can display these hand calculations instantly and automatically! We will follow todays tutorial with an example taken directly from the Hand Calculation Module in SkyCiv Beam.

Firstly, what is a bending moment? A moment is a rotational force that occurs when a force is applied perpendicularly to a point at a given distance away from that point. It is calculated as the perpendicular force multiplied by the distance from the point. A Bending Moment is simply the bend that occurs in a beam due to a moment.

It is important to remember two things when calculating bending moments; (1) the standard units are Nm and (2) clockwise bending is taken as negative. With the definitions out of the way, let’s look at the steps to calculate a bending moment diagram:

Calculating Bending Moment Diagram by Hand

1. Calculate reactions at supports and draw Free Body Diagram (FBD)

If you’re not sure how to determine the reactions at the supports – please see this tutorial first. Once you have the reactions, draw your Free Body Diagram and Shear Force Diagram underneath the beam. Finally calculating the moments can be done in the following steps:

calculate Bending Moment Diagram, bending moment of simply supported beam, moment diagram

2. From left to right, make “cuts” before and after each reaction/load

To calculate the bending moment of a beam, we must work in the same way we did for the Shear Force Diagram. Starting at x = 0 we will move across the beam and calculate the bending moment at each point.

Cut 1

Make a “cut” just after the first reaction of the beam. In our simple example:

calculate Bending Moment Diagram, bending moment of simply supported beam, moment diagram
So, when we cut the beam, we only consider the forces that are applied to the left of our cut. In this case, we have a 10kN force in the upward direction. Now as you recall, a bending moment is simply the force x distance. So as we move further from the force, the magnitude of the bending moment will increase. We can see this in our BMD. The equation for this part of our bending moment diagram is:

how to calculate the bending moment diagram of a beam, first cut

[Source: SkyCiv Beam Hand Calculation Module]

Cut 2

This cut is made just before the second force along the beam. Since there are no other loads applied between the first and second cut, the bending moment equation will remain the same. This means we can calculate the maximum bending moment (in this case at the midpoint, or x = 5) by simply substituting x=5 into the above equation:

calculate Bending Moment Diagram, bending moment of simply supported beam 

Cut 3

This cut is made just after the second force along the beam. Now we have TWO forces that act to the left of our cut: a 10kN support reaction and a -20kN downward acting load. So now we must consider both these forces as we progress along our beam. For every meter we move across the beam, there will a +10kNm moment added from the first force and -20kNm from the second.

calculate Bending Moment Diagram, bending moment of simply supported beam, moment diagram
So after the point x=5, our Bending Moment Equation becomes: M(x) = 50 +10(x-5) – 20(x-5) M(x) = 50 -10(x-5) for 5 ≤ x ≤ 10 NOTE: The reason we write (x-5) is because we want to know the distance from the pt x=5 only. Anything before this point uses a previous equation.
calculate the bending moment force in a beam hand calculation cut 2
[Source: SkyCiv Beam Hand Calculation Module]

Cut 4

Again, let’s move across to the right of our beam and make a cut just before our next force. In this case, our next cut will occur just before the reaction from Right Support. Since there are no other forces between the support and our previous cut, the equation will remain the same: M(x) = 50 -10(x-5) for 5 ≤ x≤ 10 And let’s substitute x=10 into this to find the find bending moment at the end of the beam: M(x) = 50 – 10(10-5) = 0kNm This makes perfect sense. Since our beam is static (and no rotation) it makes sense that our beam should have zero moments at this point when we consider all our forces. It also satisfies one of our initial conditions, that the sum of moments at support is equal to zero. NOTE: If your calculations lead you to any other number other than 0, you have made a mistake!

calculate Bending Moment Diagram, bending moment of simply supported beam, moment diagram
The final result, as displayed in SkyCiv Beam:
bending moment diagram example of a point load on a beam

Sign Convention for Bending Moment Diagrams

Bending Moment Diagrams can get somewhat confusing when it comes to sign conventions. You might see the same diagram drawn in completely different directions and/or positive/negative values. The two common sign conventions are displayed below, for a standard beam with a downward distributed load:

Positive Diagram with Positive Value:

The bending force is a positive force, and displayed in the positive direction. This is the convention used in the above example as we are moving from left to right along the beam:

Negative Diagram with Negative Values:

The bending force is in the negative direction, and displayed as such in the graph:

In both cases, the bending force is the same and is being generated by the sag in the beam. The important thing is that the right hand rule should govern and that you stick to the same sign convention in all parts of your design.

BONUS: How to calculate bending using SkyCiv Beam

Under our paid version of SkyCiv Beam the calculator will even show you the full hand calculations, showing the steps taken to hand calculate your bending moment diagrams. Simply model your beam using the calculator, and hit solve. It will show you the step-by-step calculations of how to draw a bending moment diagram (including cuts):

SkyCiv also has a Free Beam Calculator for you to calculate bending moment diagrams fast and easily. It also calculates reactions at supports, shear force diagrams, and deflection and span ratios. So check it out now or sign up today to get started with SkyCiv Beam!

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