고체역학 (Beer) — Pure Bending: Problem 4.93

4.93 A rectangular bar that is straight and unstressed is bent into an arc of circle of radius \(\rho\) by two couples of moment \(M\). After the couples are removed, it is observed that the radius of curvature of the bar is \(\rho_R\). Denoting by \(\rho_Y\) the radius of curvature of the bar at the onset of yield, show that the radii of curvature satisfy the following relation: \[ \frac{1}{\rho_R} = \frac{1}{\rho} \left\{ 1 - \frac{3}{2} \frac{\rho}{\rho_Y} \left[ 1 - \frac{1}{3} \left( \frac{\rho}{\rho_Y} \right)^2 \right] \right\} \]