This post explains the artificial constraints of the conjugate beam method.
Artificial Constraints
With both theorems discussed, it is more evident that the conjugate beam method is an analogy of beam analysis and an expansion of the area moment method. One advantage compared to area moment is that we don't need to draw the deflected shape to derive relationships.
However, one problem with using this method is that it obscures the theory behind the analogy.
To illustrate, let's consider cantilever beam \(AB\) as shown:
Say that we are to solve the deflection of this beam at the fixed and free ends \(A\) and \(B\) using this method. From inspection, we know that:
- The rotation and translation must be zero at the fixed end \(A\).
- The rotation and translation must not equal zero at the free end \(B\).
Now, let's construct the conjugate beam with the same support conditions (fixed end at \(A^{\prime}\) and free end at \(B^{\prime}\)). Just by viewing it, we know the following:
- Since \(A^{\prime}\) is a fixed-end support, the shear and moment at \(A^{\prime}\) must not be zero.
- Since \(B^{\prime}\) is a free end, the shear and moment must be zero.
From Theorems 1 and 2 of the conjugate beam method, you can see that there is a contradiction between the actual and conjugate structures.
For these types of beams, we have to change the support conditions of the conjugate beam to match the deflections of the actual beam. With this, you can use the first and second theorems of the conjugate beam method without the contradiction we encountered.
These changes in support are what we call artificial constraints of the conjugate beam. For simple beams, we don't need to apply such. Common examples that we need to change supports are cantilever beams, restrained beams, overhanging beams, and beams with interior supports.
The following illustration shows the conditions we must apply when creating the conjugate beam.
- We must change the fixed end of the real beam to a free one.
- We must change the free end of the real beam to a fixed one.
- We must change the simple interior supports of a real beam to an internal hinge.
- Simple exterior supports remain the same.
Summary
Let's summarize:
In constructing conjugate beams, we must change the supports of the actual beam to match the deflection behavior. These supports of the conjugate beam are what we call artificial constraints.
The rules of artificial constraints are simple: (1) we must change the fixed end of the real beam to a free one, (2) we must change the free end of the real beam to a fixed one, (3) we must change the simple interior supports of a real beam to an internal hinge. and (4) simple exterior supports remain the same.
