Conjugate beam method

The conjugate-beam methods is an engineering method to derive the slope and displacement of a beam. A conjugate beam is defined as an imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjugate beam is equal to the bending moment at that point divided by EI.^[1]

The conjugate-beam method was developed by Heinrich Müller-Breslau in 1865. Essentially, it requires the same amount of computation as the moment-area theorems to determine a beam's slope or deflection; however, this method relies only on the principles of statics, so its application will be more familiar.^[2]

The basis for the method comes from the similarity of Eq. 1 and Eq 2 to Eq 3 and Eq 4. To show this similarity, these equations are shown below.

${\displaystyle Eq.1\;{\frac {dV}{dx}}=w}$	${\displaystyle Eq.3\;{\frac {d^{2}M}{dx^{2}}}=w}$
${\displaystyle Eq.2\;{\frac {d\theta }{dx}}={\frac {M}{EI}}}$	${\displaystyle Eq.4\;{\frac {d^{2}v}{dx^{2}}}={\frac {M}{EI}}}$

Integrated, the equations look like this.

${\displaystyle V=\int w\,dx}$	${\displaystyle M=\int \left[\int w\,dx\right]dx}$
${\displaystyle \theta =\int \left({\frac {M}{EI}}\right)dx}$	${\displaystyle v=\int \left[\int \left({\frac {M}{EI}}\right)dx\right]dx}$

Here the shear V compares with the slope θ, the moment M compares with the displacement v, and the external load w compares with the M/EI diagram. Below is a shear, moment, and deflection diagram. A M/EI diagram is a moment diagram divided by the beam's Young's modulus and moment of inertia.

To make use of this comparison we will now consider a beam having the same length as the real beam, but referred here as the "conjugate beam." The conjugate beam is "loaded" with the M/EI diagram derived from the load on the real beam. From the above comparisons, we can state two theorems related to the conjugate beam:^[2]

Theorem 1: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam.

Theorem 2: The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam.^[2]