Three pinned-end columns of the same material have the same length and the same cross-sectional area (see figure). The columns are free to buckle in any direction. The columns have cross section as follows:
(1) A circle
(2) A square, and
(3) An equilateral triangle.
Determine the ratios P1 : P2 : P{3} of the critical loads for these columns.
(1)

(2)

(3)