고체역학 (Gere) — Torsion: Problem 3.3-21

3.3-21 A circular tube of inner radius \( r_1 \) and outer radius \( r_2 \) is subjected to a torque produced by forces \( P = 4000 \text{ N} \) (see figure part a). The forces have their lines of action at a distance \( b = 140 \text{ mm} \) from the outside of the tube. (a) If the allowable shear stress in the tube is \( 43 \text{ MPa} \) and the inner radius \( r_1 = 30 \text{ mm} \), what is the minimum permissible outer radius \( r_2 \)? (b) If a torsional spring of stiffness \( k_R = 50 \text{ kN} \cdot \text{m/rad} \) is added at the end of the tube (see figure part b), what is the maximum value of forces \( P \) if the allowable shear stress is not to be exceeded? Assume that the tube has a length of \( L = 450 \text{ mm} \), outer radius of \( r_2 = 37 \text{ mm} \), and shear modulus \( G = 74 \text{ GPa} \). Hint: Consider the tube and torsional spring as "springs in parallel."