Conjugate beam method

[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.

For example, as shown below, a pin or roller support at the end of the real beam provides zero displacement, but a non zero slope.

When the real beam is fixed supported, both the slope and displacement are zero.

Although this occurs, the M/EI loading will provide the necessary "equilibrium" to hold the conjugate beam stable.