The machine designer would like to calculate the allowable load for his steel column having rectangular cross section. The column has section 80 mm x 30 mm, and 380 mm long. It's proposed to use AISI 1040 hot-rolled steel. The upper end is pinned and the lower end of the column is inserted into a close-fitting socket and is welded securely as can in the picture.

To calculate the critical load for the column we need to do as following.

Solid rectangular section, 80 mm x 30 mm.

The area moment of inertia, I = 1/12 x 80 x 30

^{3}= 180000 mm

^{4}-- The least I (for this case, around X-X axis) is used.

The cross sectional area, A = 80 x 30 = 2400 mm

^{2}

Then the radius of gyration, r = (180000/2400)

^{1/2}= 8.66 mm

Since the column is pinned at the upper end and welded at the lower end, then the end fixity is fixed-pinned.

The practical value of constant K = 0.8 is used for this kind of end fixity.

The effective of the column, Le = KL = 0.8 x 380 = 304 mm

The slenderness ratio = Le/rmin = 304/8.66 = 35.1

The material AISI 1040 hot-rolled steel has yield strength (Sy) and modulus of elasticity (E) as follows.

Sy = 290 MPa

E = 207 GPa

Then we can calculate the column constant, Cc = (2p

^{2}x 207x10

^{9}/ 290x10

^{6})

^{1/2}= 118.7

Because the slenderness ratio (35.1) is less than the column constant (118.7), the column is short. The J.B. Johnson formula should be used.

The critical buckling load becomes,\

Pcr = (2400x10

^{-6}) x 290x10

^{6}[1-(290x10

^{6}x35.1

^{2})/(4p

^{2}x 207x10

^{9})] = 665571 N

With the safety factor N = 3, the allowable load for this column is

P

_{allow}= Pcr/N = 665571/3 = 221.8 kN

As usual, mechanical design handbook provides free excel program that can help you solve your column design quickly. Please download the file for FREE at the link below.

Please make sure that macro is enabled so that you can run the program. Once the program is launched, (1) select the end fixity (2) enter values of cross section, column material and safety factor (3) Click calculate to see the result.

**Download FREE Excel File of Column Design.**

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