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Conjugate beam Problem no:3 (Determining the distance a)

 Determine the distance "a" so the displacement at the edge of the beam equals the displacement at the center of the beam?

Figure 1

The conjugate beam will be, as shown in figure no:2. The pin and roller at A and B will become internal pin. The free ends will become fixed, as shown in figure no:2. We need to determine the M/EI diagram for the real beam, which will be the load in the conjugate beam, so we can determine the distance “a” that will cause an equal displacement at the end and center of the beam. 

Figure 2

Using the equilibrium equations, we will determine the reaction at point A and point B.



Figure 3



After determining the reactions at support A and B, we can now draw the shear and moment diagram. The M/EI will be the load on the conjugate beam as shown in figure no:4 and 5.

Figure 4


Figure 5


Now we need to equal the moment at the end beam and the center of the beam. we assume the distance between A and B equals L1 and L1=L-2*a.

To simplify the solution, we will take the section from point A to D. We will sum the moment at point D to determine the moment at beam end. But before that, we will take sections A, B, to determine the reaction at the internal pin (A) so we can calculate the moment at D.

So, section A-B 





Figure 6



Now. Section D-A

Now we will calculate the moment at D:





















Figure 7




Now we will take a section at the center of the beam as shown in figure no:8.


Figure 8




Now MC-C=-MD



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