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

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 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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My name is Mohammad Hamdan, and I am a Civil/bridge Engineer with extensive professional background and experience in the field of construction. I have 10+ years of experience in the construction field. During my professional career, my emphasis has been placed on structural projects, mainly bridges, infrastructure and roads. Reading and practicing the design of structures is my passion.i was lucky to be a part of a big team that executed major highway interchanges. I believe in spreading the knowledge, thus, i compose these courses to share my experience and knowledge. My course focuses on the practical side of construction field. On other word, i am sharing some of what new engineers need to know about construction field. Beside working on construction field, i love writing some articles on my website ( curious civil engineer) Moreover, i wrote some engineering papers, you can check them  by searching my name on Google (Mohammad mamon Hamdan)