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How to Calculate the Bending Moment Diagram of a Beam
Below are simple instructions on how to calculate the bending moment diagram of a simple supported beam. Study this method as it is very versatile (and can be adapted to many different types of problem. The ability to calculate the bending moment of a beam is very common practice for structural engineers and often comes up in college and high school exams.
Firstly, what is a Bending Moment? A moment is 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. Anyways, with the boring definitions out of the way, let's look at the steps to calculate a bending moment diagram:
1. Calculate reactions at supports and draw Free Body Diagram (FBD).
2. From left to right, make "cuts" before and after each reaction/load
So, when we cut the beam, we only cosider 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:
-M(x) = 10(-x)
M(x) = 50 +10(x-5) - 20(x-5)
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.
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 not rotation) it makes sense that our beam should have zero moment at this point when we consider all our forces. It also satisfies one of our initial conditions, that the sum of moments at a support is equal to zero. NOTE: If your calculations lead you to any other number other than 0, you have made a mistake!