The roof beams of a warehouse are supported by pipe columns (see figure) having outer diameter d2=100 mm and inner diameter d1= 90 mm. The columns have length L= 4.0 m, modulus E= 210 GPa, and fixed supports at the base.
Calculate the critical load Pcr of one of the columns using the following assumptions:
(1) the upper end is pinned and the beam prevents horizontal displacement;
(2) the upper end is fixed against rotation and the beam prevents horizontal displacement;
(3) the upper end is pinned but the beam is free to move horizontally; and
(4) the upper end is fixed against rotation but the beam is free to move horizontally.
