This lecture provides an overview of the slope-deflection method
for analyzing statically indeterminate beams and frames.
Similar to the force method,
the slope-deflection method is a classical structural analysis technique.
The force method uses redundant forces as unknowns to be determined
using a set of compatibility equations.
In the slope-deflection method, as the name suggests,
the unknowns are slopes and deflections at the joints of the structure-
to be calculated using a set of equilibrium equations.
Consider the continuous beam shown here:
The general deformed shape of the beam is going to look like this:
Let's refer to the beam's rotation at the supports as: theta_A, theta_B, theta_C and theta_D.
These are the unknowns to be calculated using a set of equilibrium equations.
But what are equilibrium equations, and how do we get them?
Since each joint has to be in equilibrium, sum of the moments at A, B, C and D must be 0.
So the equilibrium equations are:
To write these equations, we start by isolating the joints, like this:
Note the three beam segments:
Generally speaking, there is an internal bending moment at either end of each segment.
So for AB, we can show an internal bending moment at the left end,
and an internal bending moment at the right end of the segment.
Let's denote the left bending moment as M_AB,
and the right bending moment as M_BA.
Similarly for BC, we have M_BC at the left end,
and M_CB at the right end of the segment.
And the end moments for segment CD are M_CD and M_DC.
Needless to say, these member-end moments also show up at the joints, like this:
Since each joint has to be in equilibrium, we can write:
These are our equilibrium equations to be solved for the unknown slopes.
But the equations are not expressed in terms of the slopes,
they are expressed in terms of the member-end moments.
Therefore, we need to rewrite them in terms of theta_A, theta_B, theta_C and theta_D.
We will discuss this in details in later lectures.
For now, let's assume we are given these relations.
For this beam, they are:
Note that each member-end moment is expressed in terms of the joint rotations,
the applied load and member geometry and properties,
We call these slope-deflection equations.
Substituting the slope deflection equations in these equilibrium equations,
we get four equations in terms of the four joint rotations:
Solving the equations for the unknowns, we get:
So, here we have the joint rotations in terms of the applied load magnitude w,
member length L, and section and material properties EI.
Now that we know the joint rotations,
we can substitute them in the slope-deflection equations to get the member-end moments.
So, bending moment at the left end of segment AB is zero,
while moment at the right end of the segment is clockwise with a magnitude of wL^2/20.
For BC, we have a counterclockwise moment of wL^2/20 at the left end,
and a clockwise moment of wL^2/20 at the right end of the segment.
Segment CD has a counterclockwise moment of WL^2/20 at the left end,
and no bending moment at the right end.
We are now in a position to calculate member-end shear forces in each segment.
Here is the complete free-body diagram for segment AB.
Using these equilibrium equations, we can calculate the left-end and right-end shear forces.
We repeat this for segment BC.
Here is the segment's free-body diagram:
Here are the equilibrium equations,
and here are the member-end shear forces.
Both are 0.
And, for segment CD, we have:
Here is the free-body diagram of the three segments, showing the member-end shear forces:
Obviously, these forces also act at the joints, so we can write:
Therefore the support reactions are:
In summary, the slope-deflection method uses two sets of equations for analyzing structures.
Slope-deflection equations for relating member-end moments to joint rotations,
and joint equilibrium equations for calculating the rotations.
Once joint rotations are determined,
member-end bending moments and shear forces can be readily computed.
Central in this process is the formulation of the slope-deflection equations.
We will begin examining how these equations are obtained in the next lecture.