When analyzing statically indeterminate beams and frames using the slope-deflection method,
we often need to calculate fixed-end moments associated with applied loads.
But what are fixed-end moments?
And why exactly do we need to calculate them when using the slope-deflection method?
Recall the slope-deflection equations for a beam segment carrying a uniformly distributed load.
These terms here are called fixed-end moments.
They represent the effect of the applied load on the relationship
between the member-end moments and rotations.
So we need to have them in order to accurately model the behavior of a beam segment under loads.
Fixed-end moments also happen to represent the support reactions of a beam that is fixed at both ends.
That is, these are the support reactions for this beam.
The moment reaction at A is counterclockwise wL^2/12.
And at B, we have a clockwise moment of wL^2/12.
Although fixed-end moments for most common loading cases
are generally listed in structural analysis textbooks, or can be found online,
it would be useful to learn how to calculate them in case the need arises.
Here, we are going to present a rather straightforward strategy for calculating fixed-end moments.
First assume the beam's ends are not fixed, rather they can rotate.
So basically assume the beam is simply supported, like this:
Therefore, under the applied load, the beam is going to deflect downward, like this:
Let's denote the end rotations as theta_A and theta_B,
which we can easily calculate, say, using the virtual work method.
but we know that the real beam has a zero rotation at A and B.
So what we need is a couple of moments, one placed at A, and one placed at B,
that would undo these end rotations.
The moments force the end rotations back to zero, like this:
Put it differently, we can say that this,
equals this,
plus this:
Therefore, all we need to do here is to determine these moments,
given these known rotations.
But how do we do it?
Easily, using the slope-deflection equations for a load-free beam segment.
I'm talking about these equations which we introduced in lecture SA28.
In summary, to calculate fixed-end moments for a particular load,
first we assume the beam is simply supported and calculate its end rotations due to the load.
Then, we negate the rotations,
and plug them into these slope-deflection equations to get the fixed-end moments.
Let's illustrate this process using two examples.
Suppose we have a beam, fixed at both ends,
subjected to a downward concentrated load at mid-span.
We wish to determine its fixed-end moments.
Assume a constant EI.
To solve the problem, we first place the load on a simply supported beam and calculate its end rotations.
The rotations labeled theta_A and theta_B
can be easily computed using the virtual work method.
They are:
Then we negate the rotations.
This means the clockwise rotation at A becomes counterclockwise,
and the counterclockwise rotation at B becomes clockwise.
This gives an upward deflection to the beam, like this:
Now, plug the rotation values in the load-free slope-deflection equations
in order to determine the fixed-end moments.
Recall that these equations use our adopted sign convention,
according to which counterclockwise is considered positive.
This means, theta_A is positive PL^2/16EI.
And theta_B is negative PL^2/16EI.
Then, we get:
Now let's consider a more general loading scenario.
The beam is subjected to a concentrated load placed at an arbitrary distance
from the left fixed support, like this:
We wish to determine the beam's fixed end-moments in terms of P, L, A and B.
Assume a constant EI.
According to our strategy, we first imagined the beam is simply supported
and calculate its end rotations.
This can be done using the virtual work method.
The end rotations are:
Then, we negate the rotations to get this:
Since this is a counterclockwise rotation,
we consider it to be positive.
And, we consider this to be a negative rotation, since it is clockwise.
Now we use these slope-deflection equations to determine the fixed-end moments.
So we get:
See if you can calculate the fixed-end moments for the following loading cases: