WORLD FOR CIVIL

Topic 7.2d: Aluminum Columns - Example 4

A hollow rectangular tube is to be used as a 10-foot long vertical, fixed-fixed, column. The tube material is the aluminum alloy 2014-Ty (Alclad). The rectangular cross section has outer dimension of 4" x 8", and inner dimensions of 3" x 7", as shown in the diagram. Determine the maximum axial load that may be applied to the column before the allowable stress for buckling will be exceeded.

We first wish to determine the slenderness ratio, however we need the minimum radius of gyration, which may be calculated from r = (I/A)^1/2, where I is taken about the axis which gives the smallest value.

In this case that would be about a vertical y-y axis through the center of the cross section, resulting in the following value: I = (1/12)[b_od_o³ – b_id_i³] = (1/12)[8"*4"³ – 7"*3"³] = 26.9 in⁴. Additionally, the area = b_od_o – b_id_i = 32 in² – 21 in² = 11 in². Then r = (I/A)^1/2 = 1.56 in. And slenderness ratio = L_e/r = (.5*10' * 12"/ft)/1.56in. = 38.4

Since the slenderness ratio is greater than 12 and less then 55, we use the appropriate formula (for Alclad) for intermediate columns. The Allowable Stress = [30.7- .23(L_e/r)] ksi. = 21.87 ksi. = 21,870 lb/in².

Finally the Allowable Load = Stress * Area = 21,870 lb/in² * 11 in² = 240,570 lb.