haunched beams, and framed bents may be computed by a procedure. I. LETAL. *See H. M. Westergaard, “Deflection of Beams by the Conjugate Beam Method.

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## Conjugate beam method

Conjugate beam is defined as the 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. The conjugate-beam method was developed by H.

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 conjuhate application will be more familiar. The basis for the method comes from the similarity of Eq. To show this similarity, these equations are shown below.

Below is a shear, cnjugate, and deflection diagram. 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.

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From the above comparisons, we can state two theorems related to the conjugate beam: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam. The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam.

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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 nptsl 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.

When the real beam is fixed supported, both the slope and displacement are zero. Here the conjugate beam has a free end, since at this end there is zero shear and zero moment. Corresponding real and conjugate supports are shown below.

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Note that, as a rule, neglecting axial forces, statically determinate real beams have statically determinate conjugate beams; and statically indeterminate real beams have unstable conjugate beams. The following procedure provides a method that may be used to determine the displacement and deflection at a point on the elastic curve of a beam using the conjugate-beam method.