Question 6.EP.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. The bottom of column is fixed; the top is pinned (see Figure 6_8). Material: AISI 1040 hot-rolled steel.

Analysis       Use the procedure in Figure 6_4.

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 AISI 1040 hot-rolled steel,E = 207 GPa ands_{y} = 290 MPa. Then, from Equation (6_4),C_{c}=\sqrt{\frac{2\pi ^{2}E }{s_{y} } }

Step 3. Then KL/r < C_{c}; thus the column is short.Use the J. B. John.son formula to compute the critical load: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]                                                                                         (6_7)

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 kN.