The conjugate-beam method is an engineering method hibbeler statics pdf download derive the slope and displacement of a beam. The conjugate-beam method was developed by H. 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. Integrated, the equations look like this. Below is a shear, moment, and deflection diagram.

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.

EI diagram derived from the load on the real beam. 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.

When drawing the conjugate beam it is important that the shear and moment developed at the supports of the conjugate beam account for the corresponding slope and displacement of the real beam at its supports, a consequence of Theorems 1 and 2. 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. Consequently, from Theorems 1 and 2, the conjugate beam must be supported by a pin or a roller, since this support has zero moment but has a shear or end reaction.