Critical load of a column?

Question

I do not know how to approach this problem. I think I am making an error with the moment of inertia but I am not sure. Could anyone help me?

http://i37.photobucket.com/albums/e68/deformmechanics/column.jpg

Answer 1

Part A:

moment of inertia=I=(pi)r^4/4

Recall that the critical load for a pinned column =(pi^2EI)/L^2

For this case we multiply it by 3 since there are 3 columns acting independently (basically a third of the load is on each column so the columns can support 3 times as much).  After substituting for I we get P=(3pi^3Er^4)/4L^2

B) I am not going to go into detail on how to find the moment of inertia for this case, but just take my word that the moment of inertia of any axis through the origin is equal, thus I=5(pi)r^4/4+5(pi)r^4/4+(pi)r^4/4=11(pi)...

We use the same P equation as last time (for pin supports) and substitute for I again.  This time we do not multiply by 3 because a load on any part of the column affects the entire column.

P=(11(pi^3)Er^4)/(4L^2)

Now by doing a ratio of the two P values you see, after canceling most terms, that the column from part B allows for a critical load of ~3.67 times greater