Question 6.35: Repeat Prob. 6–34, with a completely reversed torsional mome......
Repeat Prob. 6–34, with a completely reversed torsional moment of T = 1400 lbf · in applied.
For completely reversed torsion, {k}_{a} and {k}_{b} of Prob. 6-34 apply, but {k}_{c} must also be considered.
Eq. 6-74: {k}_{c}=0.328(110)^{0.125}{{ L N}}(1,0.125) =0.590{LN}(1,0.125)
({k}_{c})_{\mathrm{torsion}}=0.328\bar{S}_{u t}^{0.125}{{L N}}(1,0.125) (6-74)
Note 0.590 is close to 0.577.
{S}_{S e}={k}_{a}\,{k}_{b}\,{k}_{c}\,{S}_{e}^{\prime}=0.768[{{ L N}}(1,0.058)](0.878)[0.590{LN}(1,0.125)][55.7{LN}(1,0.138)]\bar{S}_{Se}=0.768(0.878)(0.590)(55.7)=22.2\,\mathrm{kpsi}
C_{S e}=(0.058^{2}+0.125^{2}+0.138^{2})^{1/2}=0.195
{S}_{S e}=22.2{L}{N}(1,0.195)\;\mathrm{kpsi}
Fig. A-15-15: D/d=1.25,r/d=0.125,\,\mathrm{then}\,K_{t s}=1.40. From Eqs. (6-78), (6-79) and 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)
\mathrm{K}_{t s}={\frac{1.40{LN}(1,0.15)}{1+(2/{\sqrt{0.125}})\left[(1.4-1)/1.4\right](3/110)}}=1.34{LN}(1,0.15)\tau=\mathrm{K}_{t s}\;\frac{16T}{\pi d^{3}}
\tau=1.34[{LN}(1,0.15)]\left[\frac{16(1.4)}{\pi(1)^{3}}\right] =9.55{LN}(1,0.15)\mathrm{~kpsi}
From Eq. (5-43), p. 242:
z=-\frac{\mu_{\mathrm{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[(22.2/9.5)\sqrt{(1+0.15^{2})/(1+0.195^{2})}}{\sqrt{\ln[(1+0.195^{2})(1+0.15^{2})]}}=-3.43From Table A-10, p_{f}=0.0003
R=1-p_{f}=1-0.0003=0.9997For a design with completely-reversed torsion of 1400 lbf · in, the reliability is 0.9997. The improvement comes from a smaller stress-concentration factor in torsion. See the note at the end of the solution of Prob. 6-34 for the reason for the phraseology.
| 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 |