고체역학 (Beer) — Torsion: Problem 3.113

3.113 Three points on the nonlinear stress-strain diagram used in Prob. 3.112 are (0, 0), (0.0015, 55 MPa), and (0.003, 80 MPa). By fitting the polynomial \( \tau = A + B\gamma + C\gamma^2 \) through these points, the following approximate relation has been obtained. \[ \tau = 46.7 \times 10^9 \gamma - 6.67 \times 10^{12} \gamma^2 \] Solve Prob. 3.112 using this relation, Eq. (3.2), and Eq. (3.23). Reference from Prob. 3.112: A 50-mm-diameter cylinder is made of a brass for which the stress-strain diagram is as shown. Knowing that the angle of twist is \( 5^\circ \) in a 725-mm length, determine by approximate means the magnitude \( T \) of the torque applied to the shaft.