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Conjugate beam Problem no:1 (beam with pin and internal roller)

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 Problem no:1, Determine the slope and displacement for the following beam at point C?

Figure 1



In this example, the pin support will stay the same in the conjugate beam. While the roller will change to hinge and the free end to a fixed support. So the conjugate beam will be the same as shown in figure no:2.
Figure 2

Now we need to determine the moment diagram for the real beam. 

First step is to determine the force at support A and B by the use of equilibrium equations:

Figure 3







Now we can draw the moment diagram for the real beam. To draw the moment diagram, we can draw the shear first, and the moment diagram will be the area of shear. The moment diagram for the real beam will be, as shown in figure no:4.

Figure 4


After drawing the shear and moment diagram, our conjugate beam will be loaded with the M/EI diagram of the real beam, as shown in figure no:5.

Figure 5


now to determine slope and displacement at c we need to calculate the shear moment at c by the use of equilibrium equations:
To simplify the solution, we will divide the beam into two sections, and we will start with section A, section A showed in figure no:6. as we know, the internal hinge carries only force and no moment


Section A







































Figure 6 (section A)


Now, after determining Fb, we go to Section B to determine FC and MC.


Figure 7(section B)





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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)