Analysis of propped cantilever beams by force method

 In the video below, I explained the application of force method for analyzing indeterminate beams of type "Propped cantilever beams"

Let us see another example problem, which is slightly more difficult.

This is the example of a non-prismatic propped cantilever. Non-prismatic beam is a beam whose cross-section is not uniform throughout. In the above figure the cross-section has moment of inertia of '2I' in the first 3m and 'I' in the next 2m. Let us say the fixed end is A and the prop is provided at B.

The deflection diagram (or elastic curve) looks as shown below. Note that there is some slope at B, but deflection is zero at B. whereas at A, both deflection and slope are zero. We can conclude that the vertical distance of B (after deformation) will be zero from the tangent at A (tangent at A is horizontal line AB). This is the compatibility condition for this problem.

From Moment area theorem-II we can conclude that the moment of M/EI diagram is zero about B.For drawing the M/EI diagram we have to draw 1) M-diagram or BMD for R_B and 2) M-diagram or BMD for given loading, and then add them up. The figure below shows the M-diagram and M/EI diagram due to R_B. At the section where cross-section changes suddenly, there will be two values of M/EI diagram, one using 2I in place of I, and another using I in place of I. Note the change in shape of M/EI diagram due to non-prismatic nature of beam

Similarly, the M-diagram (BMD) due to UDL is drawn and when divided with EI it give M/EI diagram. Again, due to the non-prismatic nature, the M/EI diagram has a sharp jump at the place where I is changing.

For determining the moment of the M/EI diagrams due to R_B and UDL, we need the knowledge of areas and centroids of the figures as shown below.

Using the above areas and centroids the moment about B of the M/EI areas between A and B can be calculated as shown below

After determining R_B, other support reactions V_A and M_A can be obtained based on vertical equilibrium equation and moment equilibrium equation. By this we should get, V_A= 47.89 kN and M_A=51.75 kN-m