Old spec Statics Session 19 Example 2 (ENGR5004)

Old spec Statics Session 19 Example 2TutorialOld spec ENGR2162:210 min

ENGR216 (old spec) Statics session 19 example 2 (slide 15, statement only)

A steel column of length $L$ and rectangular cross section (sides $a$ and $b$ ) is fixed at its base B and supports a centric compressive load $P$ at its top A. Two smooth, rounded fixed plates at A stop end A from moving in the $y$ direction (so in the vertical $xy$ plane of symmetry the top is effectively pinned) but allow it to move freely in the $z$ direction (so in the $xz$ plane the top is free). Side $a$ is measured along $y$ and side $b$ along $z$ ; $x$ is the column axis.

Column of length L fixed at base B, centric load P at the top A between two smooth rounded plates; rectangular section with sides a and b; x along the column axis, y and z horizontal.

Formulas you may need

Effective lengths: fixed-free $L_e = 2L$ , fixed-pinned $L_e = 0.7L$ (on the formula sheet)
Radius of gyration $\rho = \sqrt{I/A}$ , slenderness ratio $L_e/\rho$ , $\sigma_{cr} = \dfrac{\pi^2E}{(L_e/\rho)^2}$ (on the formula sheet)
Rectangle $I = \dfrac{bh^3}{12}$ about the axis parallel to $b$ (on the formula sheet)

(a)
Determine the effective slenderness ratio for buckling in the $xy$ plane.
(b)
Determine the effective slenderness ratio for buckling in the $xz$ plane.
(c)
Determine the ratio $a/b$ of the two sides of the cross section that gives the most efficient design against buckling, and justify this choice.
a/b =-