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Conjugate beam Problem no:4 (Beam with internal pin, roller and point load )

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Example no:4: find the displacement at point C and the slope at point B for the beam shown in figure no:1.


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, to determine the displacement at point c and slope at point B.
Figure 2




To draw the moment diagram for the beam, we need to determine the reaction at point A and B, then draw the shear diagram and the moment diagram.

Using the equilibrium equations:



















Figure 3



The shear and moment diagram will be, as shown in figure no:4.


Figure 4



The conjugate beam will be loaded, as shown in figure no:5.
Figure 5



To determine the displacement at point C, we need to determine the moment in the conjugate beam at point C. to determine the moment at point C, we will take section A-B to determine the force at B.


Figure 6



Now we will take section B-C to determine the moment at point C.



Figure 7






The slope at point B equals the shear at point B in the conjugate beam. 
So



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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)