Question 6.37: Repeat Prob. 6–36, with the bar subject to a completely reve......
Repeat Prob. 6–36, with the bar subject to a completely reversed torsional moment of 2400 lbf · in.
For a non-rotating bar subjected to completely reversed torsion of T_{a}=2400\ {\mathrm{lbf}}\cdot\mathrm{in}
From Prob. 6-36:
S_{e}^{\prime}=29.3{LN}(1,0.138)\,\mathrm{kpsi}{k}_{a}=0.782{{L N}}(1,0.11)
k_{b}=0.955
For {k}_{c} use Eq. (6-74):
({k}_{c})_{\mathrm{torsion}}=0.328\bar{S}_{u t}^{0.125}{{L N}}(1,0.125) (6-74)
{k}_{c}=0.328(58)^{0.125}{{L N}}(1,0.125)=0.545{LN}(1,0.125){S}_{s e}=0.782[{LN}(1,0.11)](0.955)[0.545{LN}(1,0.125)][29.3{LN}(1,0.138)]
\bar{S}_{Se}=0.782(0.955)(0.545)(29.3)=11.9\,\mathrm{kpsi}
C_{S e}=(0.11^{2}+0.125^{2}+0.138^{2})^{1/2}=0.216
Table A-16: d/D=0,a/D=0.1,A=0.92,K_{t s}=1.68
From Eqs. (6-78), (6-79), Table 6-15
\bar{K}_{f}=\frac{K_{t}}{1+\frac{2(K_{t}-1)}{K_{t}}\frac{\sqrt{a}}{\sqrt{r}}} (6-78)
{K}_{f}=\bar{K}_{f}{LN}\left(1,C_{K_{f}}\right) (6-79)
{K}_{fs}=\frac{1.68{LN}(1,0.10)}{1+(2/\sqrt{0.125})[(1.68-1)/1.68](5/58)}=1.403{LN}(1,0.10) J_{\mathrm{net}}={\frac{\pi A D^{4}}{32}}={\frac{\pi(0.92)(1.25^{4})}{32}}=0.2201\tau_{a}={K}_{f s}\,{\frac{T_{a}c}{J_{\mathrm{net}}}}=1.403[{LN}(1,0.10)]\left[\frac{2.4(1.25/2)}{0.2201}\right] =9.56{{LN}}(1,0.10){\mathrm{~kpsi}}
From Eq. (5-43), p. 242:
z=-\frac{\mu_{{\ln} S}-\mu_{{\ln}\sigma}}{\left(\hat{\sigma}_{{\ln}S}^{2}+\hat{\sigma}_{{\ln}\sigma}^{2}\right)^{1/2}}=-\frac{\ln\left(\frac{\mu_{S}}{\mu_{\sigma}}\sqrt{\frac{1+C_{\sigma}^{2}}{1+C_{s}^{2}}}\right)}{\sqrt{\ln\left[\left(1+C_{s}^{2}\right)\left(1+C_{\sigma}^{2}\right)\right]}} (5-43)
z=-{\frac{\ln\left[(11.9/9.56){\sqrt{(1+0.10^{2})/(1+0.216^{2})}}\right]}{\sqrt{\ln[(1+0.10^{2})(1+0.216^{2})]}}}=-0.85Table A-10, p_{f}=0.1977
R=1-p_{f}=1-0.1977=0.80| Table 6–10 Parameters in Marin Surface Condition Factor |
Surface Finish | \mathrm{K}_{a}=a S_{U t}^{b}\;\mathrm{LN}(\;1,\;C) | |||
| a | b | Coefficient Variation | |||
| kpsi | Mpa | ||||
| {\mathrm{Ground}}^{*} | 1.34 | 1.58 | -0.086 | 0.120 | |
| Machined or Cold-rolled | 2.67 | 4.45 | -0.265 | 0.058 | |
| Hot-rolled | 14.5 | 58.1 | -0.719 | 0.110 | |
| As-forged | 39.8 | 271 | -0.995 | 0.145 | |
| *Due to the wide scatter in ground surface data, an alternate function is {k_{a}}=0.878{{LN}}(1,0.120). Note: S_{{Ut}} in kpsi or MPa. | |||||
| Table 6–15 Heywood’s Parameter \sqrt{a} and coefficients of variation {{C}}_{K f} for steels |
Notch Type |
{\sqrt{\alpha}}({\sqrt{\mathrm{in}}}), S_{Ut} in kpsi |
{\sqrt{\alpha}}({\sqrt{\mathrm{mm}}}),
S_{Ut} in Mpa |
Coefficient of Variation C_{KF} |
| Transverse hole | 5/{{S}}_{Ut} | 174/{{S}}_{Ut} | 0.10 | |
| Shoulder | 4/{{S}}_{Ut} | 139/{{S}}_{U t} | 0.11 | |
| Groove | 3/{{S}}_{Ut} | 104/{{S}}_{Ut} | 0.15 |