Conjugate-beam supports

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 actual beam at its supports when drawing the conjugate beam, a consequence of Theorems 1 and 2. For example, a pin or roller support at the end of the actual beam provides zero displacements but a non-zero slope, as shown below. Accordingly, a pin or roller must support the conjugate beam from Theorems 1 and 2, since this support has zero moments but has a shear or end reaction. When supporting the real beam is fixed, the slope and the displacement are both zero. The conjugate beam has a free end here since there are zero shears and zero moments to this point. Below you will find the related true and conjugate supports. Note that in general, neglecting axial forces statically determines real beams have statically determined conjugate beams, and unstable conjugate beams have statically indeterminate real beams. Even though this occurs, the M / EI loading will provide the “equilibrium” needed to keep the conjugate beam steady.

Conjugate support vs Real support

Conjugate beam method

Conjugate beam is defined as an imaginary beam of the same dimensions (length) as the original beam, but the load on the conjugate beam at any point is equal to the moment of bending at that point divided by EI. The conjugate beam method is an engineering technique for deriving the slope and displacement of a beam. H had developed the conjugate-beam process. In 1865, Müller-Breslau took over. Essentially, evaluating the slope or deflection of a beam involves the same amount of calculation as the moment-area theorems; nevertheless, this approach relies solely on the rules of statics, so its use will be more common.

The basis for the method originates from the Equations correlation. Equation 1, Equation 2, Equation 3 & Equation 4 Such calculations are shown below, in order to show this similarity.

Integrated the equations:

Procedure for Conjugate Beams

The following procedure provides a method that can be used with the conjugate-beam method to determine the displacement and deflection at a point on a beam’s elastic curve.

1. For the main spotlight, draw the conjugate projector. The beam is of the same length as the main beam and has the necessary supports as mentioned above.
2. In general, if the actual support allows a slope, the shear must be developed by the conjugate support; and if the real support allows a displacement, the conjugate support must develop for a moment.
3. The beam in the conjugate is loaded with the M / EI diagram of the real beam. It is assumed that this loading is distributed over the conjugate beam and is directed upwards when M / EI is positive and downwards when M / EI is negative. This means that the loading always acts away from the beam.
4. Using the equations of statics, determine the reactions at the conjugate beams supports.
5. Section the conjugate beam at the point where the slope θ and displacement Δ of the real beam is to be determined. In the section show the unknown shear V’ and M’ equal to θ and Δ, respectively, for the real beam.
6. In particular, if these values are positive, and the slope is counterclockwise and the displacement is upward
   Section the conjugate beam at the point where it is necessary to determine the slope and displacement of the actual beam. For the real beam, the unknown shear V’ and M’ are shown in the section equal to each and every one, respectively. In particular, if these values are positive, and the slope is counterclockwise and the displacement is upward.

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