What is Conjugate beam?
Conjugate Beam is important as 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 beams steady.
Conjugate support vs Real support
Conjugate Beam |
Real Support |
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| Fixed end | Free support | ||
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| Fixed support | Free end | ||
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| Hinged support | Hinged support | ||
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| Middle hinge | Middle support | ||
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: continue |
: continue |
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| Middle support | Middle hinge | ||
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: continue : discontinue |
: continue : discontinue |
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| Real beam | Conjugate beam | |
|---|---|---|
| Simple beam | ||
| Cantilever beam | ||
| Left-end Overhanging beam | ||
| Both-end overhanging beam | ||
| Gerber’s beam (2 spans) | ||
| Gerber’s beam (3 spans) | ||
Conjugate beam method
It 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.
For More Details: Civil engineering
Applications of Conjugate Beam
- A conjugate beam’s length is always equal to that of the actual beam.
- The M / EI representation of the loads on the main beam is the load on the conjugate beam.
- Simple support for the real beam remains simple.
- In this beam, a fixed end to the real beam becomes a free end.
- This beam is the zero shear point that corresponds to a zero slope point for the real beam.
- This beam is the maximum moment point that corresponds to a maximum deflection point for the real beam.
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.
- 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.
- 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.
- 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 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.
- Using the equations of statics, determine the reactions at the beams supports.
- Section to this beam at the point where the slope θ and displacement Δ of the real beam is to be determined. The section shows the unknown shear V’ and M’ equal to θ and Δ, respectively, for the real beam.
- In particular, if these values are positive, and the slope is counterclockwise and the displacement is upward
Section the 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 everyone, respectively. In particular, if these values are positive, and the slope is counterclockwise and the displacement is upward. Read More
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