*11-65. A narrow rectangular beam, such as shown in
the figure, can collapse when loaded through lateral
instability by twisting and displacing sidewise. It can
be shown$^{29}$ that for this case, the critical force that
may be applied at the end is
$P_{cr} = 4.013\sqrt{B_1C} / L^2$
where $B_1 = hb^3E/12$ is the flexural stiffness of the beam
around the vertical axis, and $C = \beta hb^3G$ is the tor-
sional stiffness. (For rectangular sections, coefficient
$\beta$ is given in a table in Section 4-14.)
A $5 \times \frac{1}{2}$ in narrow rectangular cantilever is made
from steel ($\sigma_{yp} = 36$ ksi and $E = 30 \times 10^3$ ksi) and is
loaded as shown in the figure. (a) Determine the critical
load $P_{cr}$ and the critical length $L_{cr}$, where both the
strength and the stability criteria are equally applica-
ble. (b) Plot $P$ vs $L$ in the neighborhood of $P_{cr}$ and $L_{cr}$
for the two criteria. (Note that the smaller of the $P$
values governs the design.)
Fig. P11-65
