Question 11.2: An 8-ft length of structural tubing has the illustrated cros......
An 8-ft length of structural tubing has the illustrated cross section and geometric properties. Using Euler’s formula and a safety factor of 2, determine the allowable centric load for the column and the corresponding normal stress. Assuming that this allowable load is applied as shown at a point 0.75 in from the geometric axis of the column, determine the horizontal deflection of the top of the column and the maximum normal stress in the column. Use E = 29 × 10^6 psi.
Given: Geometry of column, including second moment of area; safety factor.
Find: P_{cr} and σ_{cr}; w_{max} and σ_{max} if P_{cr}/2 applied eccentrically.
Assume: Hooke’s law applies.
Since the column has one fixed and one free end, the effective length is L_e = 2L = 16 ft = 192 in. Using Euler’s formula, we find the critical load to be
P_{\mathrm{cr}}=\frac{\pi^2 E I}{L_{\mathrm{e}}^2}=\frac{\pi^2\left(29 \times 10^6 \mathrm{psi}\right)\left(8.00 \mathrm{in}^4\right)}{(192 \mathrm{in})^2}=62 \mathrm{kips}Since we are asked to use a safety factor of 2, our allowable centric load is then P_{cr}/2 = 31 kips. The corresponding normal stress is
\sigma=\frac{P_{\mathrm{allow}}}{A}=\frac{31 \mathrm{kips}}{3.54 \mathrm{in}^2}=8.8 \mathrm{ksi}We have been asked for the horizontal deflection at the top of the column, which, given the supports, is the maximum deflection w_{max}. As long as we have used the correct L_e to obtain P_{cr}, we are able to use the secant formulas for eccentric loading on columns with any type of supports:
The maximum normal stress is calculated as
\begin{aligned} \sigma_{\max } & =-\frac{P}{A}\left[1+\frac{e c}{r^2} \sec \left(\frac{\pi}{2} \sqrt{\frac{P}{P_{\mathrm{cr}}}}\right)\right] \\ & =-\frac{31 \mathrm{kips}}{3.54 \mathrm{in}^2}\left[1+\frac{(0.75 \mathrm{in})(2 \mathrm{in})}{(1.50 \mathrm{in})^2} \sec \left(\frac{\pi}{2 \sqrt{2}}\right)\right] \\ & =-(8.8 \mathrm{ksi})[1+0.667(2.252)] \\ & =-22 \mathrm{ksi} \end{aligned}The sign correctly indicates that this is a compressive stress.