506 Flat Bar Bent by Couples

Problem 506
A flat steel bar, 1 inch wide by ¼ inch thick and 40 inches long, is bent by couples applied at the ends so that the midpoint deflection is 1.0 inch. Compute the stress in the bar and the magnitude of the couples. Use E = 29 × 10⁶ psi.

Solution 506

$(\rho – 1)^2 + 20^2 = \rho^2$
$(\rho^2 – 2\rho + 1) + 400 = \rho^2$
$(\rho^2 – 2\rho + 1) + 400 = \rho^2$
$2\rho = 401$
$\rho = 200.5 \, \text{in}$

$M = \dfrac{EI}{\rho}$
$f_b = \dfrac{Mc}{I} = \dfrac{(EI /\rho)c}{I}$
$f_b = \dfrac{Ec}{\rho} = \dfrac{(29 \times 10^6 )(1/8)}{200.5}$
$f_b = 18\,079.8 \, \text{psi}$
$f_b = 18.1 \, \text{ksi}$

$M = \dfrac{EI}{\rho} = \dfrac{(29 \times 10^6)\dfrac{1(1/4)^3}{12}}{200.5}$
$M = 188.3 \, \text{lb⋅in}$