how to determine the moment of inertia for different cross sectional beam like this? what is the best method to find the deflection?

beam Since the moment (and stress) is maximum at the center, the change in cross section does not make sense, at least from a strength point of view. Also the loading is symmetrical so the moment and shear also are. If the change in section is necessary for other reasons, then the smaller section must be able to handle the loads and the left side can be any size larger. The deflection curve will not be symmetrical due to the lack of symmetry of the beam.

second moment of area im in surport of proinwv view. changing the section were you have maximum bending moment will increase the deflection of the section with lower second moment of area, because it will have less flexural resistance (EI). If that sections can not sufficiently support the load the beam will fail at that section while the area with bigger EI will have little or no deflection. From design point of view it is better to have higher EI were the load will be maximum

Guys, Yes it is a lousy beam design. But can't you recognize a homework problem from a Solidmechanics course when you see one? He's not looking for improvements to the design, he just wants an answer for his homework.

Hi Asirafibrahim, Im trying to focus on your problem and I have to say I can't find the solution in terms of moment of inertia and deflection. For my opinion, the result is that you can't consider it just as a simple and single beam. The left side of the beam tends to have a smaller deflection than to the right side. At the same time the right side needs to have more deflection but this is forbidden by the left part of the beam for the principle of the continuity of the material. So you can imagine an "imaginary" double deflection curve that has a step in the middle (where is the change of section). This condition is exactly the phenomenon of the "stress concentration". Important notice: the diagrams of the moment and of the shear are the same for both side of the beam. What change are the internal stresses (shear and bending) because they depend by the area and by the inertia of the section. I hope to be clear