Estimate Kf and q for the steel shaft given in Ex. 6–6, p. 296.

Question

Accepted Answer

From Ex. 6–6, a steel shaft with [latex]S_{ut}= 690[/latex] MPa and a shoulder with a fillet of 3 mm was found to have a theoretical stress-concentration factor of [latex]K_t.= 1.65[/latex]. From
Table 6–15,

Table 6–15 Heywood’s Parameter √a and coefficients ofvariation [latex]C_{Kf}[/latex] for steels
	[latex]\sqrt{a} \left(\sqrt{in} ight) [/latex]	[latex]\sqrt{a} \left(\sqrt{mm} ight) [/latex]
Notch Type	[latex]S_{ut} \ in \ kpsi[/latex]	[latex]S_{ut} \ in MPa[/latex]	Coefficient of Variation CKf
Transverse hole	[latex]{5}/{S_{ut}}[/latex]	[latex]{174}/{S_{ut}}[/latex]	0.10
Shoulder	[latex]{4}/{S_{ut}}[/latex]	[latex]{139}/{S_{ut}}[/latex]	0.11
Groove	[latex]{3}/{S_{ut}}[/latex]	[latex]{104}/{S_{ut}}[/latex]	0.15

[latex]\sqrt{a} =\frac{139}{S_{ut}} =\frac{139}{690} =0.2014\sqrt{mm} [/latex]

From Eq. (6–78),

[latex]K_f=\frac{K_t}{1+\frac{2\left(K_t-1 ight) }{K_t} \frac{\sqrt{a} }{\sqrt{r} } } =\frac{1.65}{1+\frac{2\left(1.65-1 ight) }{1.65} \frac{0.2014 }{\sqrt{3} } } =1.51[/latex]

which is 2.5 percent lower than what was found in Ex. 6–6.

From Table 6–15, CK f = 0.11. Thus from Eq. (6–79),

[latex]K_f=\bar{K} _fLN\left(1,C_{K_f} ight) [/latex]

[latex]K_f=1.51 LN\left(1,0.11 ight) [/latex]

From Eq. (6–77), with [latex]K_t= 1.65[/latex]

[latex]\bar{q} =\frac{\bar{K}_f-1 }{K_t-1} \\hat{\sigma } _q=\frac{C\bar{K}_f}{K_t-1} \C_q=\frac{C\bar{K}_f }{\bar{K}_f-1 } [/latex]

[latex]\bar{q} =\frac{1.51-1 }{1.65-1}=0.785 \C_q=\frac{C\bar{K}_f }{\bar{K}_f-1 }=\frac{0.11\left(1.51 ight) }{1.51-1 }= 0.326\\hat{\sigma } _q=C_q\bar{q}=0.326\left(0.785 ight) =0.256[/latex]

So,

[latex]q = LN\left(0.785,0.256 ight) [/latex]

Table 6–15 Heywood’s Parameter √a and coefficients ofvariation $C_{Kf}$ for steels
	$\sqrt{a} \left(\sqrt{in} \right)$	$\sqrt{a} \left(\sqrt{mm} \right)$
Notch Type	$S_{ut} \ in \ kpsi$	$S_{ut} \ in MPa$	Coefficient of Variation CKf
Transverse hole	${5}/{S_{ut}}$	${174}/{S_{ut}}$	0.10
Shoulder	${4}/{S_{ut}}$	${139}/{S_{ut}}$	0.11
Groove	${3}/{S_{ut}}$	${104}/{S_{ut}}$	0.15

Question 2.6.18: Estimate Kf and q for the steel shaft given in Ex. 6–6, p. 2...

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