Example 5: determine the force F, so the displacement at C will be zero assuming EI constant over the beam length?

Figure 1

The first step is to draw the M/EI diagram for the real beam. The conjugate beam will be loaded with M/EI diagram for the real beam. The pin at A will stay the same for the conjugate beam while the internal roller at B will change to an internal pin for the conjugate beam. The free end at point C will become a fixed end. So the conjugate beam will be as shown in figure no:2. To draw M/EI, we need to determine A and B's reaction for the real beam. Figure 3

So the shear and moment diagram will be as shown in figure no:4.

Now, we will take section B-C. to determine FB using the equilibrium equation(MC=0).

Figure 5

Now, we will take section A-B. to determine FB using the equilibrium equation. The FB computed from section A-B will be equaled to Fb calculated from section B-C to determine F related to P.

To calculate a1 and a2, we will use the triangle similarity.

a2=a1-a

apply triangle similarities, and you will getFigure 6

Substitute a1 and a2 in the previous equation and rearrange it. You will get 3 order equation:

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