A mountain-bike rider going uphill applies torque T = Fd (F = 15 lb, d = 4 in.) 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 L1 = 2 in. and L3 = 8.5 in., and with outer diameters and thicknesses d01 = 1.25 in., t01 = 0.125 in., and d03 = 0.87 in., t03 = 0.115 in., respectively as shown. Segment BC of length L2 = 1.2 in., however, is tapered, and outer diameter and thickness vary linearly between dimensions at B and C.
Consider torsion effects only. Assume G = 4000 ksi is constant.
Derive an integral expression for the angle of twist ( D of half of the handlebar tube when it is subjected to torque T = Fd acting at the end. Evaluate D for the given numerical values.