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Conjugate-Beam method for determination of beam deflection and slope

 The Conjugate-Beam method can be used to determine the deflection and slope for beams. In this method, the shear in the conjugate beam at any point will be the real beam slope at this point. While the moment in the conjugate beam will be the displacement of the real beam at the same point. The conjugate-beam will be loaded with M/(EI), where M is the moment derived from the real beam, E is the modulus of elasticity of the beam material. I is the moment of inertia of the beam. When we draw the conjugate beam, we need to pay attention to supports. The support can be different in the conjugate beam compared to the real beam. Table no:1, show the support in the real beam and the corresponding support in the conjugate beam.

Table 1

In table 1, we can notice that the pin and roller support stay the same for the conjugate beam. The reason behind this is that the displacement at pin supports equal zero. The corresponding displacement in the conjugate beam is the moment, and the moment at pin equal zero, so pin support stay the same. On the other hand, the real beam's fixed support equals no support at the conjugate beam. As we know, the displacement and rotation at fixed support equal to zero. Therefore, if we kept the same fixed support in the conjugate beam, the moment and the shear will not be zero at this point. As we know, the moment and shear in the conjugate beam are equal to displacement and slope. Therefore if there are a moment and shear in the conjugate beam, the solution will be wrong because, at fixed support, there is no displacement or slope. As a result, we change the fixed support to free, so there will be zero moments and shear at the free end. Similarly, we can find the corresponding support for the conjugate beam. Figure no:2 shows the conjugate beams for different cases. 

Figure 2


After drawing the conjugate beam, we need to determine the moment diagram for the real beam. The conjugate beam will be loaded with the real beam M/EI diagram. after that, we can use the equilibrium equations to calculate the shear and moment at any point, which is equal to slope and displacement in the real beam. To understand this, let solve the following example.
 

Example: Determine the slope and deflection at point A for the following beam:

The first step is to draw the conjugate beam. If we go back to our table, we can notice that the corresponding support for free is fixed and for fixed is free. Therefore, the support at A will become fixed, and the support at B will become free in the conjugate beam.




The second step is to determine the loading at the conjugate beam. The load of the conjugate beam will be the real beam M/EI diagram. Therefore, we should draw the moment diagram for the real beam.

The first step is to determine the unknown force and moment at support using the equilibrium equation. 




Now we can draw the M/EI diagram



So the conjugate beam with the load will be: 




now to calculate the slope at point A, we need to determine the shear for conjugate beam at point A.




now to calculate the displacement at point A, we need to determine the moment for the conjugate beam at point A.






More examples:











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