Question 6.2: Determine the critical load on a steel column having a recta...

Objective: Compute the critical load for the column.
Given: Solid rectangular cross section: B = 12 mm; H = 18 mm; L= 280 mm.
Cross sectional area = A = BH = (12)(18) = 216 mm^2
The bottom of column is fixed; the top is pinned (see Figure 6–8).
Material: SAE 1040 hot-rolled steel.

Analysis: Use the procedure in Figure 6–7.
Results:

Step 1. Compute the slenderness ratio. The radius of gyration must be computed about the axis that gives the least value. This is the Y-Y axis, for which

r=\frac{B}{\sqrt{12}}=\frac{12 \mathrm{~mm}}{\sqrt{12}}=3.46 \mathrm{~mm}
The column has a fixed-pinned end fixity for which K=0.8. Then
K L / r=[(0.8)(280)] / 3.46=64.7
Step 2. Compute the transition slenderness ratio. For the SAE 1040 hot-rolled steel, E=207 \mathrm{GPa} and s_{y}=290 \mathrm{MPa}. Then, from Equation (6-8) (\frac{s_{y}}{2}=\frac{\pi^{2} E}{S_{r T}^{2}} \Rightarrow C_{c}=S_{r T}=\sqrt{\frac{2 \pi^{2} E}{s_{y}}}),
C_{c}=\sqrt{\frac{2 \pi^{2}\left(207 \times 10^{9} \mathrm{~Pa}\right)}{290 \times 10^{6} \mathrm{~Pa}}}=119

Step 3. Then K L/ r<C_{c}; thus the column is short. Use the J. B. Johnson formula (Eqn. 6-9) (P_{c r}=A s_{y}\left[1-\frac{s_{y} S_{r}^{2}}{4 \pi^{2} E}\right]) to compute the critical load.

\begin{aligned}&P_{c r}=A s_{y}\left[1-\frac{s_{y}(K L/ r)^{2}}{4 \pi^{2} E}\right] \\&P_{c r}=\left(216 \mathrm{~mm}^{2}\right)\left(290 \mathrm{~N} / \mathrm{mm}^{2}\right)\left[1-\frac{\left(290 \times 10^{6} \mathrm{~Pa}\right)(64.7)^{2}}{4 \pi^{2}\left(207\times 10^{9} \mathrm{~Pa}\right)}\right] \\&P_{c r}=53.3 \times 10^{3}\mathrm{~N}=53.3 \mathrm{kN}\end{aligned}
Comments: This is the critical buckling load. We would have to apply a design factor to determine the allowable load. Specifying N=3 results in P_{a}=17.8 \mathrm{kN}.