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

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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