This post explains the first theorem of the conjugate beam method.
The first theorem deals with finding rotation \(\theta\) at a specific point. It states that: the fictional shear of the conjugate beam at any point is the rotation (slope) of the actual beam at said point.
\(V^{\prime}=\theta\)
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Let's focus on the figure shown to verify this theorem. We have a simple beam with length \(L\) loaded with uniformly distributed load \(w\). In addition, we draw its deflected shape with deviation \(t_{B/A}\).
Suppose that we are interested in finding the rotation at \(A\). From the beam's deflected shape, we can say it is:
Equation 1: \(\theta_A=\frac{t_{B/A}}{L}\)
Below the actual beam is the fictional loading \(\frac{M}{EI}\) of the conjugate beam. For this example, we use moment by parts (with a section cut at \(B\)) to compute its area and centroid later on efficiently.
Let's compute for the shear at the left end \(A^{\prime}\) - the parallel point of \(A\) in the real beam. According to the theorem, this value must be equal to Equation 1. To solve for \(R_{A^{\prime}}\), we take a summation of moments at \(B^{\prime}\), the right end:
\(\sum{M_B^{\prime}}=0]\circlearrowright_+\)
\(R_{A^{\prime}}\left( L \right)+\frac{1}{3}\left( L \right)\left( \frac{wL^2}{2EI} \right)\left( \frac{L}{4} \right)-\frac{1}{2}\left( L \right)\left( \frac{wL^2}{2EI} \right)\left( \frac{L}{3} \right)=0\)
\(R_{A^{\prime}}\left( L \right)=\frac{L}{2}\left( \frac{wL^2}{2EI} \right)\left( \frac{L}{3} \right)-\frac{L}{3}\left( \frac{wL^2}{2EI} \right)\left( \frac{L}{4} \right)\)
This result is the expression we use to solve for the reaction at \(A^{\prime}\). Look closely at the right side of the equation. It is equal to the first moment of area about \(B'\), which is equal to deviation \(t_{B/A}\). Therefore, we can say that:
\(R_{A^{\prime}}\left( L \right)=t_{B/A}\)
Equation 2: \(R_{A^{\prime}}=\frac{t_{B/A}}{L}\)
Equations 1 and 2 are equal - it shows the validity of the first theorem. To stress out again, if you take the shear \(V^{\prime}\) of the conjugate beam at a point, you are solving for the rotation \(\theta\) of the actual beam at said point.
Let's summarize:
The fictional shear of the conjugate beam at any point is the rotation (slope) of the actual beam at said point.
\(V^{\prime}=\theta\)
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