Question

Determine the critical load, Per, required to cause failure of a 21 ft long column made of A-36 structural steel with a moment of inertia of 2.19 in* about the y-y axis, 16.4 in about the x-x axis, and a cross sectional area of A = 2.68 in2. Assume that the column behaves as if it is fixed at its base, and fixed at the top. Be sure to check both buckling possibilities, and the crushing failure mode when determining the failure load.

Answer 1

Given Data:

Length (L) = 21 ft.

MOI (Ix) = 16.4 in2

MOI (Iy) = 2.19 in4

Area (A) = 2.68 in2

Assumption:

The column is fixed at both ends, Thus n = 4.

The elastic modulus for A-36 steel is 29,000,000 psi

Solution:

Critical Buckling load K = (1/4)^0.5 = 0.5

Applying Euler's Formula for buckling on the xx-axis.

$Pcr = \frac{\pi ^2 E I}{(KL)^2}$

$Pcr = \frac{\pi ^2 (29) (10)^6 (16.4)}{((0.5)(252))^2}$

Pcr = 295.67 kpi (Buckling in xx-axis)

Applying Euler's Formula for buckling on the yy-axis.

$Pcr = \frac{\pi ^2 E I}{(KL)^2}$

$Pcr = \frac{\pi ^2 (29) (10)^6 (2.19)}{((0.5)(252))^2}$

Pcr = 394.82 kpi   (Buckling in yy-axis)

$\sigma cr = \frac{Pcr}{A}$

= 110.32 ksi (for xx-axis)

= 147.32 ksi   (for yy-axis)