고체역학 (Gere) — Torsion: Problem 3.4-17

3.4-17 A mountain-bike rider going uphill applies torque \( T = Fd \) (\( F = 65 \text{ N}, d = 100 \text{ mm} \)) to the end of the handlebars \( ABCD \) by pulling on the handlebar extenders \( DE \). Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at \( A \)). Segments \( AB \) and \( CD \) are prismatic with lengths \( L_1 = 50 \text{ mm} \) and \( L_3 = 210 \text{ mm} \) and with outer diameters and thicknesses \( d_{01} = 40 \text{ mm}, t_{01} = 3 \text{ mm} \) and \( d_{03} = 22 \text{ mm}, t_{03} = 2.8 \text{ mm} \) respectively as shown. Segment \( BC \) of length \( L_2 = 38 \text{ mm} \) however, is tapered, and outer diameter and thickness vary linearly between dimensions at \( B \) and \( C \). Consider torsion effects only. Assume \( G = 28 \text{ GPa} \) is constant. Derive an integral expression for the angle of twist \( \phi_D \) of half of the handlebar tube when it is subjected to torque \( T = Fd \) acting at the end. Evaluate \( \phi_D \) for the given numerical values.