Bending Moment is mostly used to find bending stress. It is the internal torque holding a beam together (stopping the left and right halves from rotating - if it was to break in half!)
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Definition of Bending Moment
Bending Moment is the torque that keeps a beam together (anywhere along the beam).
It is found by cutting the beam, then calculating the MOMENT needed to hold the left (or right) half of the beam stationary.
If this is done for the other (left) side you should get the same answer - but opposite direction.
Bending Moment in 19 seconds...
In the video above, the wooden plank has been cut through at mid span. The only thing holding it together is the spring loaded hinge. This hinge applies a clockwise moment (torque) to the RHS, and a counter-clockwise moment to the LHS.
The hinge is applying a moment to BOTH sides of the beam. This is called Bending Moment. You can't normally see it happening unless the beam breaks, but bending moment is being applied everywhere along the length of the beam.
The gymnast pushes each arm downwards - hard. He is applying a moment to each arm, turning himself into a "beam" between each ring. The longer the arms the greater the bending moment - which is why the wrist is turned inwards, slightly reducing the length of each arm.
Positive Bending Moment
This type of bending is common - where the load is pushing down and reactions at the end push upwards. This is called positive bending.
In a more strict sense, positive bending is a sagging beam.
Positive bending is whenever the beam tends to sag downwards. Negative bending bows upwards - called hogging.
To calculate the Bending Moment at any location along the beam, we "cut" the beam at that point, then do a moment equation for ONE SIDE of the beam (left or right - whichever is easier). You don't do both sides because the moments balance each other and you will get ZERO. (Because it is equilibrium - of course)
We can cut the beam to find the Bending Moment (by doing moment equation of ONE side).But what if we don't know where the maximum is? Are we going to pick lots of places to "cut" until we eventually find it?
There is another way. We can use shear force to find bending moment, using a diagram method.
But first - a definition of shear force in a beam.
Imagine ONLY the sliding aspect of the beam, not the bending. This is a bit wierd since beams don't slide apart, they bend apart. But anyway, we still need to get this shear force thing happening.
Imagine the beam as a stack of magnets. They can slide OK but not bend.
Now if one hand pushes up and the other down, it slides. (shears)
Of course, it doesn't matter which magnet slides, they all want to slide. This one just happened to have the least friction, but in fact every slice has the same shear force.
Positive Shear Force
In a diagram we would show it like this;
When the left hand side (LHS) goes up then this is called positive shear force.
A Shear Force Diagram is a graph of the shear force all the way along a beam.
Consider the Bending Moment at Points A,B,C and D on the beam below;
Shear Force Diagram
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Assignment: Questions 28:1 and 28:2, p360-361 (Chapter 28: Shear force and Bending Moment)