Shear and Moment Diagrams
Previously we talked about shear and moment in beams, and how to represent them using algebraic equations.
In this session, let's focus on the graphical representation of shear and moments,
also called shear and moment diagrams.
Using these diagrams, we can quickly and accurately determine where in the beam
shear and bending stresses are maximum or minimum,
and how the stresses vary across the beam,
an ability that plays an important role in structural design.
Before we get to explain how shear and moment diagrams can be drawn,
let's establish a sign convention.
In this example, note that the left region of the shear diagram is labeled positive,
and the diagram's right region is labeled negative.
What exactly does positive or negative shear mean?
That is what the sign convention clarifies.
Consider a simple beam.
If we cut it, say at point C,
we need to show the internal shear force and bending moment at the cut-point.
The shear force has to be shown as a pair of vertical forces acting in opposite directions,
so the pair has to be drawn either like this,
or like this:
In the first scenario the shear force tends to rotate the beam's segments in the clockwise direction.
In the second scenario the rotation of the beam's segments tend to be in the counterclockwise direction.
By convention, the shear force direction that tends to cause a clockwise rotation is considered positive.
If the shear force tends to cause a counterclockwise rotation, we call it a negative shear.
Therefore this is a positive shear:
And this is a negative shear:
Similarly, when we cut the beam, the internal bending moment is drawn either like this,
or like this:
In the first scenario, the bending moment tends to cause a concave up deformation in the beam, like this:
In the second scenario, the beam's segments tend to deform concave down like this:
A positive moment tends to cause a concave up bending in the beam,
a negative moment tends to bend the beam concave down,
therefore this is considered a positive moment:
And this is a negative moment:
Now on to drawing shear and moment diagrams.
One way to draw the diagrams is to graph their corresponding equations.
For example, for this beam we can easily formulate the shear and moment equations.
We talked about the procedure for coming up with these equations in a previous session,
so we're not going to talk about that again today.
Let's just assume they have been derived already.
Here they are:
Now we want to graph them.
For shear we have two equations.
For the left segment of the beam, where X is between 0 and 5, shear equals to: 375-100X
This represents a line segment, to draw it we need two points,
say, the segment's endpoints.
So we evaluate the equation at: X=0 and X=5
At X=0, the equation gives us 375.
At X=5, we get -125
First locate the points on the coordinate system, then connect them with a straight line.
Note that the line crosses the X axis between 0 and 5,
making the left part of the diagram positive, and the right part negative.
For X between 5 and 10, shear is given using the equation -125
Since here shear is not really a function of X, it remains constant throughout the segment.
So now we have the complete shear diagram for the beam.
We can graph the moment equation in a similar manner.
For the first segment of the beam where X is between 0 and 5,
the moment equation is quadratic.
Therefore we need more than two points to graph it.
I'm going to use three points for generating the graph,
the segment's end points and the point where moment is maximum.
How do we determine the point of maximum?
When moment is maximum, the slope of the equation is going to be 0.
This means at the point of maximum the derivative of the moment vanishes.
That is, since the moment equation is: M=375X-50X^2, then dM/dX must be 0.
That is, dM/dX=375-100X=0
Solving the equation for X, we get: X=3.75
That is, moment is maximum at X=3.75
So now we have three points for graphing the moment equation.
At X=0, we have: M=0
At X=3.75, we have M=703
And, at X=5, M=625
Here is the graph of the quadratic equation:
Since the diagram is above the X axis, moment is considered positive in the region between 0 and 5.
Moment in the right segment of the beam where X is between 5 and 10,
is given by the equation: 1250-125X
This is a line equation, to graph it, let's evaluate the equation at its endpoints.
At X=5, we get: 625
At X=10, we get: 0
So moment changes linearly from 625 to 0 in this segment, like this:
In summary for this beam,
this is the shear diagram,
and this is the moment diagram.
There is a technique for drawing shear and moment diagrams without using any equations.
We will discuss and illustrate that technique in the next session.