A hollow steel shaft is to tr mit 4200 N ・ m of torque and is to be sized so that the torsional
stress does not exceed 120 MPa.
(a) If the inside diameter is 70 percent of the outside diameter, what size shaft should be used?
Use preferred sizes.
(b) What is the stress on the inside of the shaft when full torque is applied?
(a)
\begin{aligned}&\tau=\frac{T c}{J} \Rightarrow 120\left(10^{6}\right)=\frac{4200(d / 2)}{(\pi / 32)\left[d^{4}-(0.70 d)^{4}\right]}=\frac{2.8149\left(10^{4}\right)}{d^{3}} \\&d=\left(\frac{2.8149\left(10^{4}\right)}{120\left(10^{6}\right)}\right)^{1 / 3}=6.17\left(10^{-2}\right) \mathrm{m}=61.7 \mathrm{~mm}\end{aligned}
From Table A-17 the next preferred size is d=80 \mathrm{~mm} . . d_{i}=0.7 d=56 \mathrm{~mm} . The next preferred size smaller is d_{i}=50 \mathrm{~mm} .
(b)
\tau=\frac{T c}{J}=\frac{4200\left(d_{i} / 2\right)}{(\pi / 32)\left[d^{4}-\left(d_{i}\right)^{4}\right]}=\frac{4200(0.050 / 2)}{(\pi / 32)\left[(0.080)^{4}-(0.050)^{4}\right]}=30.8 \mathrm{MPa} \quad \text { }table A-17
| A17 |
| Fraction of Inches |
| \frac{1}{64}, \frac{1}{32}, \frac{1}{16}, \frac{3}{32}, \frac{1}{8}, \frac{5}{32}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \frac{3}{8}, \frac{7}{16}, \frac{1}{2}, \frac{9}{16}, \frac{5}{8}, \frac{11}{16}, \frac{3}{4}, \frac{7}{8}, 1,1 \frac{1}{4}, 1 \frac{1}{2}, 1 \frac{3}{4}, 2,2 \frac{1}{4}, 2 \frac{1}{2}, 2 \frac{3}{4}, 3, 3 \frac{1}{4}, 3 \frac{1}{2}, 3 \frac{3}{4}, 4,4 \frac{1}{4}, 4 \frac{1}{2}, 4 \frac{3}{4}, 5,5 \frac{1}{4}, 5 \frac{1}{2}, 5 \frac{3}{4}, 6,6 \frac{1}{2}, 7,7 \frac{1}{2}, 8,8 \frac{1}{2}, 9,9 \frac{1}{2}, 10,10 \frac{1}{2}, 11,11 \frac{1}{2}, 12, 12 \frac{1}{2}, 13,13 \frac{1}{2}, 14,14 \frac{1}{2}, 15,15 \frac{1}{2}, 16,16 \frac{1}{2}, 17,17 \frac{1}{2}, 18,18 \frac{1}{2}, 19,19 \frac{1}{2}, 20 |
| Decimal Inches |
| 0.010, 0.012, 0.016, 0.020, 0.025, 0.032, 0.040, 0.05, 0.06, 0.08, 0.10, 0.12,\\ 0.16, 0.20, 0.24, 0.30, 0.40, 0.50, 0.60, 0.80, 1.00, 1.20, 1.40, 1.60, 1.80, 2.0, 2.4, 2.6,\\ 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, 6.0, 7.0,\\ 7.5, 8.5, 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, 12.0, 12.5, 13.0, 13.5, 14.0, 14.5, 15.0, 15.5, 16.0,\\ 16.5, 17.0, 17.5, 18.0, 18.5, 19.0, 19.5, 20 |
| Millimeters |
| 0.05, 0.06, 0.08, 0.10, 0.12, 0.16, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80,\\ 0.90, 1.0, 1.1, 1.2, 1.4, 1.5, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5,\\ 6.0, 6.5, 7.0, 8.0, 9.0, 10, 11, 12, 14, 16, 18, 20, 22, 25, 28, 30, 32, 35, 40, 45, 50,\\ 60, 80, 100, 120, 140, 160, 180, 200, 250, 300 |
| Renard Numbers* |
| 1st choice, R5: 1, 1.6, 2.5, 4, 6.3, 10 |
| 2d choice, R10: 1.25, 2, 3.15, 5, 8 |
| 3d choice, R20: 1.12, 1.4, 1.8, 2.24, 2.8, 3.55, 4.5, 5.6, 7.1, 9 |
| 4th choice, R40: 1.06, 1.18, 1.32, 1.5, 1.7, 1.9, 2.12, 2.36, 2.65, 3, 3.35, 3.75, 4.25, 4.75, 5.3, 6, |
| 6.7, 7.5, 8.5, 9.5 |