Lets make it a bit more comprehesive by using the same geometry for both the conditions, i.e. static and then to dynamic.. Since I was not using the same geometry for my previous two examples, it might have become a bit difficult to understand. Observe the first condition below.
As already discussed, this type of boundary condition is not exactly the same as assumed by the general analytical equations while finding out the max stresses. The flexural formula assumes the fixed boundary condition is applied on the complete cross section at the other end, but lets just consider the above example for now. For initial sizing phase, assume flexural formula can be used to find out the max stresses and then the necessary analytic sizing process is conducted to re-define the size of this structure in order to optimize it for weight. [We haven't reached the FEA yet, just initial sizing].
CONDITION 2: (Applied force is a follower force)
Now how can I size this structure using analytical equations? The structure is flexible in reality, but I don't know how to size this structure to optimize it for weight. [Again, no FEA is involved initially while sizing].