Question 6.EP.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: AISI 1020 hot-rolled steel.

Analysis        Use the procedure in Figure 6_13. Assume first that the column is long.

Results         From Equation (6_9),D=\left[\frac{64 N P_{a}(K L)^{2}}{\pi^{3} E}\right]^{1 / 4}

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 AISI 1020 hot-rolled steel,s_{y}= 30 000 psi. The graph in Figure 6_5 showsC_{c}to be approximately 138. Thus, the actual KL/r is less than the transition value, and the column must be redesigned as a short column, using Equation (6_10) derived from the Johnson formula:D=\left[\frac{4 N P_{a}}{\pi s_{y}}+\frac{4 s_{y}(K L)^{2}}{\pi^{2} E}\right]^{1 / 2}

                                                                                            (6_10)

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.