Question 6-3: Specify a suitable diameter of a solid, round cross section ...

Objective: Specify a suitable diameter for the column.
Given: Solid circular cross section: L = 25 in; use N = 3.
Both ends are pinned.
Material: SAE 1020 hot-rolled steel.
Analysis: Use the procedure in Figure 6–13. Assume first that the column is long.
Results: From Equation (6–11),

\begin{aligned}&D=\left[\frac{64 N P_{a}(K L)^{2}}{\pi^{3} E}\right]^{1 / 4}=\left[\frac{64(3)(9800)(25)^{2}}{\pi^{3}\left(30 \times 10^{6}\right)}\right]^{1 / 4} \\&D=1.06 \text { in }\end{aligned}
The radius of gyration can now be found:
r=D / 4=1.06 / 4=0.265 \mathrm{in}
The slenderness ratio is
K L / r=[(1.0)(25)] / 0.265=94.3
For the SAE 1020 hot-rolled steel, s_{y}=30000 psi. The graph in Figure 6-5 shows C_{c} to be approximately 138. Thus, the actual K L /r is less than the transition value, and the column must be redesigned as a short column, using Equation (6-12) derived from the Johnson formula:
\begin{aligned}&D=\left[\frac{4 N P_{a}}{\pi s_{y}}+\frac{4 s_{y}(K L)^{2}}{\pi^{2} E}\right]^{1 / 2} \\&D=\left[\frac{4(3)(9800)}{(\pi)(30000)}+\frac{4(30000)(25)^{2}}{\pi^{2}\left(30 \times 10^{6}\right)}\right]^{1 / 2}=1.23 \mathrm{in}\end{aligned}
Checking the slenderness ratio again, we have
K L /r=[(1.0)(25)] /(1.23 / 4)=81.3

Comments: This is still less than the transition value, so our analysis is acceptable. A preferred size of D = 1.25 in could be specified.
An alternate method of using spreadsheets to design columns is to use an analysis approach similar to that shown in Figure 6–9 but to use it as a convenient “trial and error” tool. You could compute data by hand, or you could look them up in a table for A, I, and r for any desired cross-sectional shape and dimensions and insert the values into the spreadsheet. Then you could compare the computed allowable load with the required value and choose smaller or larger sections to bring the computed value close to the required value. Many iterations could be completed in a short amount of time. For shapes that allow computing r and A fairly simply, you could add a new section to the spreadsheet to calculate these values. An example is shown in Figure 6–14, where the differently shaded box shows the calculations for the properties of a round cross section. The data are from Example Problem 6–3, and the result shown was arrived at with only four iterations.